Digital infrastructure strategies: the case of the province of Caserta

Paolo Pane; Federico de Andreis; Paolo Pane; Federico de Andreis

doi:10.3934/geosci.2023014

AIMS Geosciences

2023, Volume 9, Issue 2: 243-257. doi: 10.3934/geosci.2023014

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Research article

Digital infrastructure strategies: the case of the province of Caserta

Paolo Pane ^{1
,
,},
Federico de Andreis ^{2
,
,}

1.
Dipartimento di Scienze politiche, Università di Napoli Federico Ⅱ, Italy
2.
Scienze e tecnologie dei trasporti, Università Giustino Fortunato, Italy

Received: 12 September 2022 Revised: 17 February 2023 Accepted: 12 March 2023 Published: 31 March 2023

The digital economy and the associated productivity gains generated by the diffusion of the Internet are considered fundamental components of growth models. Scientific reflection converges in considering balanced access to digital services as a diriment factor for the promotion of competitiveness, equity, economic development and social and environmental sustainability. Although the availability of infrastructure is not sufficient to achieve the full development of the territory and the community, it is nevertheless an unavoidable prerequisite for today's and future technological and digital applications and, therefore, investigating the type of association between the presence of communication networks and the socio-economic structure of the territories is essential to understanding the very nature of multidimensional inequalities and their spatial and geographical distribution, within a framework that sees infrastructure as a conversion factor and means of development for capabilities. Based on the theories of social exclusion, the capability approach and critical theory, the research presented aims to investigate, through the analysis of a case study, the possible association between the state of progress of broadband implementation and specific territorial configurations, considering also different variables of a geographical nature.

Keywords:

Citation: Paolo Pane, Federico de Andreis. Digital infrastructure strategies: the case of the province of Caserta[J]. AIMS Geosciences, 2023, 9(2): 243-257. doi: 10.3934/geosci.2023014

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Abstract

1. Introduction

Nonlinear oscillators have been widely used in various engineering and applied sciences, such as mathematics, physics, structural dynamics, mechanical engineering and other related fields of science ^{[1,2,3,4,5,6]}. Nonlinear differential equations (NDEs) can model many phenomena in various scientific aspects to present their effects and behaviors through mathematical principles. Perturbation methods are extremely beneficial when the nonlinear response is small ^[7,8,9,10]. In general, solving strongly nonlinear differential equations is very difficult. In ^[11], Mickens suggested an approximate expression for solving a truly nonlinear Duffing oscillator. Recently, various powerful analytical and numerical approximation techniques have been suggested for dealing with nonlinear oscillator differential equations. These include He's frequency-amplitude formulation ^[12], the harmonic balance method ^[13,14], the straightforward frequency prediction method ^[15], the modified harmonic balance method ^[16,17], the energy balance method ^[18,19,20], the homotopy perturbation method ^{[21,22,23,24,25]}, the Hamiltonian approach ^[26,27], the weighted averaging method ^[28], the global residue harmonic balance method ^[29,30,31], the max-min approach ^[32,33], Newton's harmonic balance method ^[34], the variational iteration method ^[35], the parameter-expansion method ^[36], the Lindstedt-Poincaré method ^[37,38] and the global error minimization method ^{[39,40,41,42]}.

The global error minimization method (GEMM) is one of the most frequently used techniques for dealing with the solutions of nonlinear oscillators, as it provides more accurate results valid for both weakly and strongly nonlinear oscillators than other known methods ^{[39,40,41,42]}. In the previous example, the solution up to the first approximation is calculated.

In this study, we extend and improve the global error minimization method up to a third order approximation to achieve the higher-order analytical solution of strongly nonlinear Duffing-harmonic oscillators. The present method is applied for two different problems, and the analytical results show that the modified global error minimization method (MGEMM) has better agreement with numerical solutions than the other analytical methods. Excellent agreement is observed between the approximate and exact solutions even for large amplitudes of the oscillation. Comparing the exact solutions with the approximate results has proved that the MGEMM is quite an accurate method in strongly nonlinear oscillator systems.

2. Basic idea of the modified global error minimization method (MGEMM)

To describe the proposed modified global error minimization method (MGEMM), we consider general second order nonlinear oscillator differential equations as follows:

$\ddot u + F(\dot u, \, u, \, t) = 0, \, \;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.\;\;\;\;$

(1)

We introducing $E(u)$ as a new function, defined as in ^[40,43], in the following form:

$E(u) = \int\nolimits_0^T \;{\left( {\ddot u + F(\dot u, \, u, \, t)} \right)^2}dt, \, \;\;\;\;\;T = 2\pi {\omega ^{ - 1}}.$

(2)

By assuming that $F(u)$ is an odd function, a general n-th order trial function of Eq (1) can be expressed as a sum of trigonometric functions as follows:

$u(t) = \mathop \sum \limits_{n = 0}^\infty ({a_{2n + 1)}}\cos ((2n + 1)\omega t)),$

(3)

where ${a_{(2n + 1)}}$ are unknown constant values which satisfy the relation

$A = \mathop \sum \limits_{n = 0}^\infty {a_{(2n + 1)}}.$

(4)

The following conditions were used to obtain the unknown parameters (i.e., ${a_{(2n + 1)}}$ and $\omega$ ):

$\frac{{\partial E(u)}}{{\partial \omega }} = 0, \, \;\;\;\;\;\;\;\;\frac{{\partial E(u)}}{{\partial {a_{(2n + 1)}}}} = 0, \, \;\;\;n \geqslant 1.\;\;$

(5)

By solving the n Eq (5) with the aid of Eq (4), the constants ${a_1}, \, \, {a_3}, {a_5}$ and the frequency of vibration $\omega$ are obtained.

3. Examples of two nonlinear Duffing-harmonic oscillators

In this section, two practical examples of nonlinear Duffing-harmonic oscillators are illustrated to show the effectiveness, accuracy and applicability of the proposed approach.

3.1. Example 1

In this application, we consider the following nonlinear Duffing-harmonic oscillator:

$\ddot u + {k_1}u + \frac{{{k_3}u}}{{1 + {k_2}{u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0, \,$

(6)

where dots denote differentiation with respect to $t$ . Now, we shall study some different relevant cases considering Eq (6).

3.1.1. Case 1

First, we consider ${k_1} = 1$ , ${k_2} = 1$ and ${k_3} = 1$ in Eq (6). Then, we have a nonlinear oscillator system having an irrational elastic item ^[12,16].

$\ddot u + u + \frac{u}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\;\;\dot u(0) = 0.$

(7)

According to the basic idea of the global error minimization method, the minimization problem of Eq (7) is

$E(u) = \int\nolimits_0^T {\left( {\ddot u + u + \frac{u}{{1 + {u^2}}}} \right)^2}dt, \, \;\;\;T = 2\pi /\omega .$

(8)

The first-order approximate solution for Eq (7) can be represented as a trial function in the form

${u_1}(t) = {a_1}\cos (\omega t).$

(9)

Substituting Eq (9) into Eq (8) and choosing ${a_1} = A,$ it follows that

$E({u_1}) = \frac{{4{A^2}\pi }}{\omega } + \frac{{3{A^4}\pi }}{\omega } + \frac{{5{A^6}\pi }}{{8\omega }} - 4{A^2}\pi \omega - \frac{9}{2}{A^4}\pi \omega - \frac{5}{4}{A^6}\pi \omega + {A^2}\pi {\omega ^3} + \frac{3}{2}{A^4}\pi {\omega ^3} + \frac{5}{8}{A^6}\pi {\omega ^3} = 0.$

(10)

Applying $\partial E({u_1})/\partial \omega = 0$ , the frequency of the nonlinear oscillator is obtained as follows:

$\omega = {\omega _1} = \sqrt {\frac{{16 + 18{A^2} + 5{A^4} + 2\sqrt {256 + 576{A^2} + 487{A^4} + 180{A^6} + 25{A^8}} }}{{3(8 + 12{A^2} + 5{A^4})}}} .$

(11)

In order to illustrate the capacity of the global error minimization method, the second-order approximation is applied to the Duffing-harmonic oscillator by using the following new trial solution:

${u_2}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t), \,$

(12)

where $A = {a_1} + {a_3}.$ By substituting Eq (12) into Eq (8), we have

$\left. {\begin{array}{*{20}{l}} {E({u_2}) = \tfrac{\pi }{{8\omega }}\left( {5{{({\omega ^2} - 1)}^2}a_1^6 + 5(11{\omega ^4} - 14{\omega ^2} + 3)a_1^5{a_3} + 2a_1^3{a_3}(8(2 - 9{\omega ^2} + 5{\omega ^4})} \right.} \\ { + \left. {3(5 - 50{\omega ^2} + 109{\omega ^4})a_3^2} \right) + 3a_1^4\left( {4(2 - 3{\omega ^2} + {\omega ^4}) + (15 - 110{\omega ^2} + 159{\omega ^4})a_3^2} \right)} \\ { + a_3^2\left( {8{{(2 - 9{\omega ^2})}^2} + 12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^4} \right)} \\ { + \left. {a_1^2\left( {8{{( - 2 + {\omega ^2})}^2} + 16(6 - 45{\omega ^2} + 59{\omega ^4})a_3^2 + 3(15 - 190{\omega ^2} + 559{\omega ^4})a_3^4} \right)} \right) = 0.} \end{array}} \right)\,$

(13)

Setting $\partial E({u_2})/\partial \omega = 0$ and $\partial E({u_2})/\partial {a_3} = 0$ leads to

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}(20\omega ( - 1 + {\omega ^2})a_1^6 + 5( - 28\omega + 44{\omega ^3})a_1^5{a_3} + 2a_1^3{a_3}(8( - 18\omega + 20{\omega ^3}) + 3( - 100\omega + 436{\omega ^3})a_3^2} \\ { + 3a_1^4(4( - 6\omega + 4{\omega ^3}) + ( - 220\omega + 636{\omega ^3})a_3^2) + a_3^2( - 288\omega (2 - 9{\omega ^2}) + 12( - 54\omega + 324{\omega ^3})a_3^2} \\ { - 180\omega (1 - 9{\omega ^2})a_3^4) + a_1^2(32\omega ( - 2 + {\omega ^2}) + 16( - 90\omega + 236{\omega ^3})a_3^2 + 3( - 380\omega + 2236{\omega ^3})a_3^4))} \\ { - \tfrac{1}{{8{\omega ^2}}}\pi (5{{( - 1 + {\omega ^2})}^2}a_1^6 + 5(3 - 14{\omega ^2} + 11{\omega ^4})a_1^5{a_3} + 2a_1^3{a_3}(8(2 - 9{\omega ^2} + 5{\omega ^4}) + 3\left( {5 - 50{\omega ^2}} \right.} \\ {\left. { + 109{\omega ^4}} \right)\, a_3^2) + 3a_1^4(4(2 - 3{\omega ^2} + {\omega ^4}) + (15 - 110{\omega ^2} + 159{\omega ^4})a_3^2) + a_3^2(8{{(2 - 9{\omega ^2})}^2}} \\ { + 12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^4) + a_1^2(8{{( - 2 + {\omega ^2})}^2} + 16(6 - 45{\omega ^2} + 59{\omega ^4})a_3^2} \\ { + 3(15 - 190{\omega ^2} + 559{\omega ^4})a_3^4)) = 0, } \end{array}} \right)\,$

(14)

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}(5(3 - 14{\omega ^2} + 11{\omega ^4})a_1^5 + 6(15 - 110{\omega ^2} + 159{\omega ^4})a_1^4{a_3} + 12(5 - 50{\omega ^2} + 109{\omega ^4})a_1^3a_3^2} \\ { + 2a_1^3\left( {8(2 - 9{\omega ^2} + 5{\omega ^4}) + 3(5 - 50{\omega ^2} + 109{\omega ^4})a_3^2} \right) + a_3^2\left( {24(2 - 27{\omega ^2}81{\omega ^4}){a_3} \\+ 20{{(1 - 9{\omega ^2})}^2}a_3^3} \right)} \\ { + a_1^2\left( {32(6 - 45{\omega ^2} + 59{\omega ^4}){a_3} + 12(15 - 190{\omega ^2} + 559{\omega ^4})a_3^3} \right) + 2{a_3}\left( {8{{(2 - 9{\omega ^2})}^2}} \right.} \\ { + \left. {12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^2)} \right) = 0.} \end{array}} \right)\,$

(15)

For a known amplitude, the parameters of ${a_1}$ , ${a_3}$ and angular frequency $\omega$ can be obtained by using the condition $A = {a_1} + {a_3}$ and solving Eqs (14) and (15). The computations were performed using the Mathematica software program, version 9.

To illustrate the capacity of this method, the third order approximation is applied by using the following trial function:

${u_3}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t) + {a_5}\cos (5\omega t), \,$

(16)

where $A = {a_1} + {a_3} + {a_5}$ . Bringing Eq (16) into Eq (8) results in

$\left. \begin{gathered} \begin{array}{*{20}{l}} {E({u_3}) = \, \tfrac{\pi }{{8\omega }}\left( {5a_1^6} \right.{{\left( {{\omega ^2} - 1} \right)}^2} + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 5a_3^6{{\left( {1 - 9{\omega ^2}} \right)}^2}} \\ { + 2a_1^3{a_3}\left( {3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + a_5^2\left( {5a_5^4{{\left( {1 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) \\+ 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \end{array} \hfill \\ \begin{array}{*{20}{l}} { + a_3^2{{\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8\left( {2 - 9{\omega ^2}} \right){\omega ^2}} \right)}^2}} \\ {\left. {\left. { + 3a_5^2\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 18{a_5}a_3^3\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right)} \right.} \\ {\left. { + 8{{\left( {{\omega ^2} - 2} \right)}^2}} \right) + 9a_5^4\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \\ {4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right)\,$

(17)

Applying $\partial E({u_3})/\partial \omega = 0$ , $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ yields

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {20a_1^6\omega \left( {{\omega ^2} - 1} \right) + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {110{a_3}\omega + 54{a_5}\omega } \right) + 2a_1^5\omega \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right)} \right.} \\ {\left. { + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 2a_1^3{a_3}\left( {3a_3^2\left( {436{\omega ^3} - 100\omega } \right) + 3{a_5}{a_3}\left( {2652{\omega ^3} - 460\omega } \right) + 8\left( {20{\omega ^3} - 18\omega } \right)} \right.} \\ {\left. { + 10a_5^2\left( {1004{\omega ^3} - 124\omega } \right)} \right) + 3a_3^4\left( {a_5^2\left( {11644{\omega ^3} - 860\omega } \right) + 4\left( {324{\omega ^3} - 54\omega } \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {11576{\omega ^3} - 1240\omega } \right) + 15{a_5}{a_3}\left( {612{\omega ^3} - 52\omega } \right) + 8\left( {1148{\omega ^3} - 198\omega } \right)} \right.} \\ {\left. { + 3a_5^2\left( {13596{\omega ^3} - 940\omega } \right)} \right) + a_5^2\left( {12a_5^2\left( {2500{\omega ^3} - 150\omega } \right) - 800\omega \left( {2 - 25{\omega ^2}} \right)} \right.} \\ {\left. { - 500a_5^4\omega \left( {1 - 25{\omega ^3}} \right)} \right) + a_3^2\left( {3a_5^4\left( {22524{\omega ^3} - 1180\omega } \right) + 16a_5^2\left( {3212{\omega ^3} - 306\omega } \right) \\- 288\omega \left( {2 - 9{\omega ^2}} \right)} \right)} \\ \begin{gathered} + a_1^4\left( {a_3^2\left( {1908{\omega ^3} - 660\omega } \right) + 4{a_5}{a_3}\left( {1468{\omega ^3} - 380\omega } \right) + 3\left( {3a_5^2\left( {1108{\omega ^3} - 180\omega } \right) \\+ 4\left( {4{\omega ^3} - 6\omega } \right)} \right)} \right) \hfill \\ \begin{array}{*{20}{l}} {a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 380\omega } \right) + 18{a_5}a_3^3\left( {1364{\omega ^3} - 180\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) \\+ 32\omega \left( {{\omega ^2} - 2} \right)} \right.} \\ { + 9a_5^4\left( {5268{\omega ^3} - 340\omega } \right) + 4a_3^2\left( {3a_5^2\left( {6972{\omega ^3} - 700\omega } \right) + 4\left( {236{\omega ^3} - 90\omega } \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {10876{\omega ^3} - 860\omega } \right) + 8\left( {196{\omega ^3} - 54\omega } \right)} \right)} \right)} \right)} \\ { - \tfrac{\pi }{{8{\omega ^2}}}\left( {5a_1^6{{\left( {{\omega ^2} - 1} \right)}^2} + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 5a_3^6{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \\ { + 2a_1^3{a_3}\left( {3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) \\+ 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + a_5^2\left( {5a_3^4{{\left( {1 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) \\+ 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ {\left. {\left. { + 3a_5^2\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 8{{\left( {{\omega ^2} - 2} \right)}^2}} \right.} \\ { + 9a_5^4\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right) + 18a_3^3{a_5}\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right)} \\ { + 4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \end{array}} \right)\,$

(18)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\, \tfrac{\pi }{{8\omega }}\left( {5a_1^5} \right.\left( {{\omega ^2} - 1} \right)\, \left( {11{\omega ^2} - 3} \right) + 30a_3^5{{\left( {1 - 9{\omega ^2}} \right)}^2} + 2{a_1}a_3^2{a_5}\left( {2{a_3}\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right)} \right.} \\ {\left. { + 15{a_5}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right)} \right) + a_1^4\left( {2{a_3}\left( {477{\omega ^4} - 330{\omega ^2} + 45} \right) + 4{a_5}\left( {367{\omega ^4} - 190{\omega ^2} + 15} \right)} \right)} \\ { + 2a_1^3{a_3}\left( {6{a_3}\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right)} \right) + 2a_1^3\left( {8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right)} \\ { + 12a_3^3\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right) + \\ 4{a_1}{a_3}{a_5}\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right)} \\ { + 2{a_3}\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ { + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 54a_3^2{a_5}\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right) + } \right.\, 8{a_3}\left( {4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right.} \\ {\left. {\left. {\left. { + 3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right)} \right) + 6{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + \\ 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(19)

$\begin{gathered} \left. \begin{gathered} \begin{array}{*{20}{l}} {\, \tfrac{\pi }{{8\omega }}\left( {3a_1^5} \right.\left( {{\omega ^2} - 1} \right)\, \left( {9{\omega ^2} - 1} \right) + 6a_3^4{a_5}\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 2a_1^3{a_3}\left( {3{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right)} \right.} \\ {\left. { + 20{a_5}\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + a_1^4\left( {4{a_3}\left( {367{\omega ^4} - 190{\omega ^2} + 15} \right) + 18{a_5}\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {15{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 6{a_5}\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + \\ 2{a_1}a_3^2\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ { + \left. {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right)} \\ { + a_5^2\left( {20a_5^3{{\left( {1 - 25{\omega ^2}} \right)}^2} + 24{a_5}\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right)} \right) + a_3^2\left( {32{a_5}\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 12a_5^3\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right)} \right) + 2{a_5}\left( {12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right.} \\ {\left. { + 5a_5^4{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {18a_3^3\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right) + 96{a_5}\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \right.} \\ { + 36a_5^3\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 12{a_3}a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 24a_3^2{a_5} \\\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right)} \\ {\left. {\left. { + 6{a_3}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right) \hfill \\ \, \hfill \\ \end{gathered}$

(20)

Now, by solving Eqs (18)–(20) and applying the condition $A = {a_1} + {a_3} + {a_5}$ , the parameters ${a_1}, \, {a_3}, \, {a_5}$ and the angular frequency $\omega$ can be obtained for the known amplitude $A,$ using the Mathematica software program, version 9. To examine the accuracy of the MGEMM solutions, the obtained results are compared with the frequency-amplitude formulation (FAF) ^[12], the energy balance method (EBM) ^[19], the modified harmonic balance method (MHBM) ^[16] and the exact solutions, as presented in Table 1 and Figure 1. We conclude that the third order approximation provides an excellent accuracy with respect to the exact numerical solutions.

Table 1. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{FAF}}$	${\omega _{EBM}}$	${\omega _{MHBM}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[12]	^[19]	^[16]	present	Exact
0.01	1.41419	1.41419	1.41419	1.41419	1.41419
0.1	1.41158	1.41158	1.41158	1.41158	1.41158
0.2	1.40388	1.40389	1.40390	1.40390	1.40390
0.4	1.37581	1.37595	1.37616	1.37616	1.37616
0.6	1.33694	1.33743	1.33827	1.33827	1.33827
0.8	1.29448	1.29550	1.29744	1.29743	1.29743
1	1.25375	1.25514	1.25845	1.25840	1.25842
5	1.02500	1.02588	1.03148	1.02945	1.03139
10	1.00656	1.00681	1.00895	1.00790	1.00893
100	1.00007	1.0000	1.00010	1.00010	1.00010
1000	1.00000	1.00000	1.00000	1.00000	1.00000

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Figure 1. Comparison of the approximate solution (red line) with the numerical solution (blue line).

DownLoad: Full-Size Img PowerPoint

3.1.2. Case 2

Now, if we put ${k_1} = 0$ , ${k_2} = 1$ and ${k_3} = 1$ in Eq (6), we obtain the following nonlinear oscillator, in which the restoring force has a rational expression ^[21,26].

$\ddot u + \frac{u}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.$

(21)

Using the previously mentioned procedure, the solution up to a third-order approximation is calculated. Depending on the analytical approximation, first, second or third, the approximate solution is assumed in the forms of (9), (12) and (16), respectively.

Finally, as in Case 1, the third order approximate solutions are compared with the homotopy perturbation method (HPM) ^[21], the Hamiltonian approach (HA) ^[26] and the exact solutions, as displayed in Table 2.

Table 2. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{HPM}}$	${\omega _{HA}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[21]	^[26]	present	Exact
0.01	0.999963	0.9999625	0.999963	0.99999
0.1	0.996271	0.99627403	0.996273	0.9991208
1	0.755929	0.765366864	0.761539	0.76157808
10	0.114708	0.13420106	0.11948	0.123322
100	0.0115462	0.01407125	0.0120357	0.0125265

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3.2. Example 2

In the second application, we will consider the following nonlinear Duffing-harmonic oscillator ^[18]:

$\ddot u + {k_1}u + \frac{{{k_3}{u^3}}}{{1 + {k_2}{u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\;\;\dot u(0) = 0, \,$

(22)

where dots denote differentiation with respect to $t$ . Now, we consider some following cases to compare the present solutions with published solutions using different approximate analytical methods.

3.2.1. Case 1

First, we consider ${k_1} = 1,$ ${k_2} = 1$ and ${k_3} = 1$ in Eq (22). Then, we have the following nonlinear Duffing-harmonic equation ^[18,28]:

$\ddot u + u + \frac{{{u^3}}}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \, \, \dot u(0) = 0.$

(23)

The minimization problem is

$E(u) = \int\nolimits_0^T {\left( {\ddot u + u + \frac{{{u^3}}}{{1 + {u^2}}}} \right)^2}dt, \, \;\;T = \frac{{2\pi }}{\omega }.$

(24)

For the first-order approximation, assume that the trial function is given by

${u_1}(t) = {a_1}\cos (\omega t), \,$

(25)

where $A = {a_1}$ . By inserting Eq (25) into Eq (24), we obtain

$E({u_1}) = \frac{{{A^2}\pi }}{\omega } + \frac{{3{A^4}\pi }}{\omega } + \frac{{5{A^6}\pi }}{{2\omega }} - 2{A^2}\pi \omega - \frac{9}{2}{A^4}\pi \omega - \frac{5}{2}{A^6}\pi \omega + {A^2}\pi {\omega ^3} + \frac{3}{2}{A^4}\pi {\omega ^3} + \frac{5}{8}{A^6}\pi {\omega ^3} = 0.$

(26)

The frequency can be found through the condition $\partial E({u_1})/\partial \omega = 0$ , as follows:

$\omega = {\omega _1} = \sqrt {\frac{{8 + 18{A^2} + 10{A^4} + 2\sqrt {64 + 288{A^2} + 487{A^4} + 360{A^6} + 100{A^8}} }}{{3(8 + 12{A^2} + 5{A^4})}}} .$

(27)

To improve the analytical approximation, we add additional terms to the trial function:

${u_2}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t).$

(28)

The constraint of this minimization is $A = {a_1} + {a_3}$ . Substituting the above new trial function into Eq (24), we obtain

$\left. {\begin{array}{*{20}{l}} {E({u_2}) = \, \tfrac{\pi }{{8\omega }}\left( {5a_1^6} \right.{{\left( {{\omega ^2} - 2} \right)}^2} + 5a_1^5{a_3}\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 3a_1^4\left( {4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right.} \\ {\left. { + a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \\ { + a_3^2\left( {5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2} + 12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right)} \\ {\left. { + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 16a_3^2\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right)} \right) = 0.} \end{array}} \right)\,$

(29)

By using $\partial E({u_2})/\partial \omega = 0$ and $\partial E({u_2})/\partial {a_3} = 0$ , it follows that

$\left. {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {5{a_3}a_1^5} \right.\, \left( {44{\omega ^3} - 56\omega } \right) + 3a_1^4\left( {a_3^2\left( {636{\omega ^3} - 440\omega } \right) + 4\left( {4{\omega ^3} - 6\omega } \right)} \right) + 20a_1^6\omega \left( {{\omega ^2} - 2} \right)} \end{array}} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {1308{\omega ^3} - 600\omega } \right) + 8\left( {20{\omega ^3} - 18\omega } \right)} \right) + a_3^2\left( {12a_3^2\left( {324{\omega ^3} - 54\omega } \right) - 288\omega \left( {1 - 9{\omega ^3}} \right)} \right.} \\ {\left. {\left. { - 180a_3^4\omega \left( {2 - 9{\omega ^3}} \right)} \right) + a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 760\omega } \right) + 16a_3^2\left( {236{\omega ^3} - 90\omega } \right) + 32\omega \left( {{\omega ^2} - 1} \right)} \right)} \right)} \\ { - \tfrac{\pi }{{8{\omega ^2}}}\left( {5a_1^6{{\left( {{\omega ^2} - 2} \right)}^2} + 5{a_3}a_1^5\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 3a_1^4\left( {a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right)} \right.} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right) + a_3^2\left( {12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \\ {\left. {\left. { + 5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 16a_3^2\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right)} \right) = 0, } \end{array}} \right)\,$

(30)

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {5a_1^5\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 6{a_3}a_1^4\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4a_3^2a_1^3\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right)} \right.} \\ { + 2a_1^3\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right) + a_3^2\left( {24{a_3}\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 20a_3^3{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 32{a_3}\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. { + 2{a_3}\left( {5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2} + 12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right)} \right) = 0.} \end{array}} \right)\,$

(31)

The minimization problem's conditions can be easily achieved by replacing ${a_1} = A - {a_3}$ , and the parameters ${a_1}$ , ${a_3}$ and angular frequency $\omega$ can be obtained for a known amplitude $A$ .

To show the accuracy of the MGEM method in higher order approximations, we apply the third order approximation and consider the following trial function:

${u_3}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t) + {a_5}\cos (5\omega t).$

(32)

Using Eq (32) as the trial function in Eq (24), where $A = {a_1} + {a_3} + {a_5}$ , leads to

$\left. {\begin{array}{*{20}{l}} {E({u_3}) = \, \tfrac{{\pi \omega }}{8}\left( {5a_1^6} \right.\left( {{\omega ^2} - 4} \right) + a_1^5\left( {5{a_3}\left( {11{\omega ^2} - 28} \right) + 3{a_5}\left( {9{\omega ^2} - 20} \right)} \right) + 45a_3^6\left( {9{\omega ^2} - 4} \right)} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^2} - 300} \right) + 10a_5^2\left( {251{\omega ^2} - 124} \right) + 3{a_3}{a_5}\left( {663{\omega ^2} - 460} \right) + 40{\omega ^2}} \right)} \\ { + a_1^4\left( {a_3^2\left( {477{\omega ^2} - 660} \right) + 9a_5^2\left( {277{\omega ^2} - 180} \right) + 4{a_3}{a_5}\left( {367{\omega ^2} - 380} \right) + 12{\omega ^2}} \right)} \\ { + 3a_3^4\left( {a_5^2\left( {2911{\omega ^2} - 860} \right) + 324{\omega ^2}} \right) + 2{a_1}a_3^2{a_5}\left( {2a_3^2\left( {1447{\omega ^2} - 620} \right) + 2296{\omega ^2}} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^2} - 940} \right) + 15{a_3}{a_5}\left( {153{\omega ^2} - 52} \right)} \right) + 125a_5^2\left( {60a_5^2{\omega ^2} + 40{\omega ^2}} \right.} \\ {\left. { + a_5^4\left( {25{\omega ^2} - 4} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^2} - 380} \right) + 944a_3^2{\omega ^2} + 8{\omega ^2}} \right.} \\ { + 6{a_3}{a_5}\left( {3a_3^2\left( {341{\omega ^2} - 180} \right) + 392{\omega ^2}} \right) + 12a_5^2\left( {7a_3^2\left( {249{\omega ^2} - 100} \right) + 484{\omega ^2}} \right)} \\ {\left. { + 9a_5^4\left( {1317{\omega ^2} - 340} \right) + 6{a_3}a_5^3\left( {2719{\omega ^2} - 860} \right)} \right) + a_3^2\left( {12848a_5^2{\omega ^2} + 648{\omega ^2}} \right.} \\ {\left. {\left. { + 3a_5^4\left( {5631{\omega ^2} - 1180} \right)} \right)} \right) = 0.} \end{array}} \right)\,$

(33)

By setting $\partial E({u_3})/\partial \omega = 0$ , $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ , we obtain

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{1}{{8\omega }}\pi \left( {20a_1^6\omega } \right.\left( {{\omega ^2} - 2} \right) - 180a_3^6\omega \left( {2 - 9{\omega ^2}} \right) + a_1^5\left( {{\omega ^2} - 2} \right)\, \left( {110{a_3}\omega + 54{a_5}\omega } \right) + 2{\omega _3}a_1^5\left( {5{a_3}\left( {11{\omega ^2} - 6} \right)} \right.} \\ {\left. { + 3{a_5}\left( {9{\omega ^2} - 2} \right)} \right) + 3a_3^4\left( {a_5^2\left( {11644{\omega ^3} - 1720\omega } \right) + 4\left( {324{\omega ^3} - 54\omega } \right)} \right) + 2{a_1}a_3^2{a_5}\left( {8\left( {1148{\omega ^3} - 198\omega } \right)} \right.} \\ {\left. { + 2a_3^2\left( {5788{\omega ^3} - 1240\omega } \right) + 15{a_5}{a_3}\left( {612{\omega ^3} - 104\omega } \right) + 3a_5^2\left( {13596{\omega ^3} - 1880\omega } \right)} \right) + a_5^2\left( { - 800\omega \left( {1 - 25{\omega ^2}} \right)} \right.} \\ {\left. { + 12a_5^2\left( {2500{\omega ^3} - 150\omega } \right) - 500a_5^4\omega \left( {2 - 25{\omega ^2}} \right)} \right) + a_3^2\left( {16a_5^2\left( {3212{\omega ^3} - 306\omega } \right) - 288\omega \left( {1 - 9{\omega ^2}} \right)} \right.} \\ {\left. { + 3a_5^4\left( {22524{\omega ^3} - 2360\omega } \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {1308{\omega ^3} - 600\omega } \right) + 3{a_5}{a_3}\left( {2652{\omega ^3} - 920\omega } \right)} \right.} \\ {\left. { + 2\left( {5a_5^2\left( {1004{\omega ^3} - 248\omega } \right) + 4\left( {20{\omega ^3} - 18\omega } \right)} \right)} \right) + a_1^4\left( {3a_3^2\left( {636{\omega ^3} - 440\omega } \right) + 4{a_5}{a_3}\left( {1468{\omega ^3} - 760\omega } \right)} \right.} \\ {\left. { + 3\left( {a_5^2\left( {3324{\omega ^3} - 1080\omega } \right) + 4\left( {4{\omega ^3} - 6\omega } \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 760\omega } \right) + 32\omega \left( {{\omega ^2} - 1} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} { + 18a_3^3{a_5}\left( {1364{\omega ^3} - 360\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) + 9a_5^4\left( {5268{\omega ^3} - 680\omega } \right) + 4a_3^2\left( {4\left( {236{\omega ^3} - 90\omega } \right)} \right.} \\ { + 18a_3^3{a_5}\left( {1364{\omega ^3} - 360\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) + 9a_5^4\left( {5268{\omega ^3} - 680\omega } \right) + 4a_3^2\left( {4\left( {236{\omega ^3} - 90\omega } \right)} \right.} \\ {\left. { - \tfrac{{1 - \pi }}{{8{\omega ^2}}}} \right)\, 5a_1^6{{\left( {{\omega ^2} - 2} \right)}^2} + a_1^5\left( {{\omega ^2} - 2} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 6} \right) + 3{a_5}\left( {9{\omega ^2} - 2} \right)} \right) + 5a_3^6{{\left( {2 - 9{\omega ^2}} \right)}^2}} \\ { + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right) + 2{a_1}a_3^2{a_5}\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right)} \\ { + a_5^2\left( {5a_5^4{{\left( {2 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right) + a_3^2\left( {8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right. + } \\ {\left. {3a_5^4\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3{a_3}{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right) + 2\left( {5a_5^2\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right) + 4\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \right)} \\ { + a_1^4\left( {3a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4{a_5}{a_3}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right)} \right. + 3\left( {4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right.} \\ {\left. {\left. {a_5^2 + \left( {831{\omega ^4} - 540{\omega ^2} + 60} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right.} \\ { + 9a_5^4\left( {1317{\omega ^4} - 340{\omega ^2} + 20} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right) + 18a_3^3{a_5}\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right)} \\ { + 4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right) + 6{a_3}{a_5}\left( {8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. {\left. {\left. { + a_5^2 + \left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(34)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\, \left( {3a_1^5} \right.\left( {{\omega ^2} - 2} \right)\, \left( {9{\omega ^2} - 2} \right) + 6a_3^4{a_5}\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right) + 2a_1^3{a_3}\left( {3{a_3}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 20{a_5}\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right)} \right) + a_1^4\left( {4{a_3}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right) + 6{a_5}\left( {831{\omega ^4} - 540{\omega ^2} + 60} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {15{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 6{a_5}\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right) + 2{a_1}a_3^2\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right)} \\ { + a_5^2\left( {20a_5^3{{\left( {2 - 25{\omega ^2}} \right)}^2} + 24{a_5}\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right)} \right) + a_3^2\left( {32{a_5}\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 12a_5^3\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right)} \right) + 2{a_5}\left( {12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right.} \\ {\left. { + 5a_5^4{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {18a_3^3\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right) + 96{a_5}\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \right.} \\ { + 24{a_5}a_3^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right) + 12a_5^2{a_3}\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right)} \\ {\left. {\left. { + 36a_5^3\left( {1317{\omega ^4} - 340{\omega ^2} + 20} \right) + 6{a_3}\left( {a_5^2\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(35)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{1}{{8\omega }}\pi \left( {\, 5a_1^5} \right.\left( {{\omega ^2} - 2} \right)\, \left( {11{\omega ^2} - 6} \right) + 30a_3^5{{\left( {2 - 9{\omega ^2}} \right)}^2} + 2{a_1}a_3^2{a_5}\left( {4{a_3}\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 15{a_5}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right)} \right) + a_1^4\left( {6{a_3}\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4{a_5}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right)} \right)} \\ { + 2a_1^3{a_3}\left( {2{a_3}\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 3{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right)} \right) + 12a_3^3\left( {4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. { + a_5^2\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right)} \right) + 4{a_1}{a_3}{a_5}\left( {2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right) + 15{a_3}{a_5}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right)} \right) + 2{a_3}\left( {8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3a_5^4\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right) + 2a_1^3\left( {a_3^2\left( {327\omega _3^4 - 300{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 3{a_3}{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right) + 2\left( {5a_5^2\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right) + 4\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \right)} \\ { + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 54a_3^2{a_5}\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right) + } \right.\, 8{a_3}\left( {4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right.} \\ {\left. {\left. {\left. { + 3a_5^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right)} \right) + 6{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right)\,$

(36)

Putting ${a_1} = A - {a_3} - {a_5}$ in the minimization problem changes the constraint minimization problem to an unconstrained minimization problem, which is easier to find the solution of Eq (33) by using the conditions $\partial E({u_3})/\partial \omega = 0,$ $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ .

We plot the analytical solutions obtained from MGEMM (red line) Eq (32) and compare them with numerical solutions of Eq (23) obtained using the fourth order Runge-Kutta method (blue line). It is observed that for all different values of the amplitude A, the approximate solutions match extremely well with the numerical solutions (see Figure 2).

Figure 2. Comparison of the approximate solution (red line) with the numerical solution (blue line).

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3.2.2. Case 2

Second, we consider ${k_1} = 0,$ ${k_2} = 1$ and ${k_3} = 1$ in Eq (6). Hence, we have the one-dimensional nonlinear oscillator governed by ^{[21,22,23,24,25,26]}

$\ddot u + \frac{{{u^3}}}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.$

(37)

Finally, we can obtain the first and second-order approximations to Eq (37) given by Eqs (25) and (28), respectively. We remark that the third-order approximation in Eq (32) can be given by MGEMM in a similar manner. Generally, after three steps of MGEMM, one can obtain the approximated solutions to Eq (37) with sufficient accuracy. The analytical results of Eq (37) are compared with the iterative homotopy harmonic balance method (IHHBM) ^[34], the energy balance method (EBM) ^[20], the max-min approach (MMA) ^[32], the global residue harmonic balance method (GRHBM) ^[29], the Hamiltonian approach (HA) ^[26] and the exact solutions, as shown in Table 3.

Table 3. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{EBM}}$	${\omega _{IHHBM}}$	${\omega _{MMA}}$	${\omega _{GRHBM}}$	${\omega _{HA}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[20]	^[34]	^[32]	^[29]	^[26]	present	Exact
0.01	0.00866	0.008478	0.00866	0.008472	0.00865	0.00847	0.00847
0.1	0.08627	0.084418	0.08627	0.084394	0.08624	0.08439	0.08439
1	0.65164	0.63136	0.65465	0.636795	0.64359	0.636783	0.63678
5	0.97343	0.96667	0.97435	0.968107	0.96731	0.969202	0.96698
10	0.99314	0.99090	0.99340	0.991591	0.99095	0.992005	0.99092
50	0.99973	0.99961	0.99973	0.999657	0.999608	0.999676	0.99961
100	0.99999	0.999901	0.99993	0.999914	0.99990	0.999919	0.99990

| Show Table

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4. Results and discussion

In this paper, we test the analytical solutions of strongly nonlinear Duffing-harmonic oscillators to show the effectiveness of MGEMM. Comparisons of the analytical solutions and the exact numerical solutions of the Duffing-harmonic oscillators for small and large values of the amplitude have been illustrated in Figures 1 and 2 and Tables 1–3. In these Figures, the analytical solution is indicated by a red line, while the numerical solution is represented by a blue line. There is good compatibility between the analytical and numerical solutions, which confirms the accuracy of our results, and excellent matching is observed in these calculations. Moreover, as shown in Tables 1–3, the results produced using the MGEMM are in better agreement with those obtained using exact solutions than other existing ones in the literature. The calculations of the above applications were done using Mathematica software.

5. Conclusions

In this paper, the modified global error minimization method has been presented successfully to obtain higher order approximate periodic solutions of strongly nonlinear Duffing-harmonic oscillators. The present method, which is proved to be a powerful mathematical tool to study nonlinear oscillators, can be easily extended to any nonlinear equation because of its efficiency and convenient applicability. We demonstrated the accuracy and efficiency of the proposed method by solving some examples. We showed that the obtained solutions are valid for the whole domain. Comparisons of the obtained solutions, whether numerical or analytical, revealed a clear match, highlighting the precision of the modified GEMM.

Acknowledgments

M. Zayed and G. M. Ismail extend their appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through the research groups program under grant R.G.P.2/207/43.

Conflict of interest

The authors declare that they have no conflict of interest.

References

[1]	Helpman E (1998) Introduction. In General purpose technologies and economic growth, Cambridge: MIT Press, 1–14.
[2]	Malecki EJ (2002) The economic geography of the Internet's infrastructure. Econ Geogr 78: 399–424. https://doi.org/10.2307/4140796 doi: 10.2307/4140796
[3]	Doyle E, Perez-Alaniz M (2017) From the Concept to the Measurement of Sustainable Competitiveness: Social and Environmental Aspects. Entrepreneurial Bus Econ Rev 5: 35–59. https://doi.org/10.15678/EBER.2017.050402 doi: 10.15678/EBER.2017.050402
[4]	European Commission, 8th Cohesion Report, 2022. Available from: https://ec.europa.eu/regional_policy/en/information/publications/communications/2022/cohesion-in-europe-towards-2050-8th-cohesion-report.
[5]	Rekowski M, Piekarz T, Sztokfisz B, et al. (2020) International competition in the digital age, Geopolitics of emerging and disruptive technologies, Krakow: The Kosciuszko Institute, 13–24.
[6]	Costa JDS, Ellson RW, Martin RC (1987) Public capital, regional output and development: some empirical evidence. J Regional Sci 27: 419–437. https://doi.org/10.1111/j.1467-9787.1987.tb01171.x doi: 10.1111/j.1467-9787.1987.tb01171.x
[7]	Holtz-Eakin D (1994) Public sector capital and the productivity puzzle. Rev Econ Stat 76: 12–21. https://doi.org/10.2307/2109822 doi: 10.2307/2109822
[8]	Holtz-Eakin D, Schwartz AE (1995) Spatial productivity spillovers from public infrastructure: evidence from state highways. Int Tax Public Finan 2: 459–468. https://doi.org/10.1007/BF00872777 doi: 10.1007/BF00872777
[9]	Boarnet MG (1998) Spillovers and the locational effects of public infrastructure. J Regional Sci 38: 381–400. https://doi.org/10.1111/0022-4146.00099 doi: 10.1111/0022-4146.00099
[10]	Sloboda BW, Yao VW (2008) Interstate spillovers of private capital and public spending. Ann Reg Sci 42: 505–518. https://doi.org/10.1007/s00168-007-0181-z doi: 10.1007/s00168-007-0181-z
[11]	Crescenzi R, Rodríguez-Pose A (2012) Infrastructure and regional growth in the European Union. Pap Reg Sci 91: 487–615. https://doi.org/10.1111/j.1435-5957.2012.00439.x doi: 10.1111/j.1435-5957.2012.00439.x
[12]	Borowiecki R, Siuta-Tokarska B, Maroń J, et al. (2021) Developing Digital Economy and Society in the Light of the Issue of Digital Convergence of the Markets in the European Union Countries. Energies 14: 2717. https://doi.org/10.3390/en14092717 doi: 10.3390/en14092717
[13]	Seitz H, Licht G (1995) The impact of public infrastructure capital on regional manufacturing production cost. Reg Stud 29: 231–240. https://doi.org/10.1080/00343409512331348923 doi: 10.1080/00343409512331348923
[14]	Moomaw RL, Mullen JK, Williams M (1995) The interregional impact of infrastructure capital. South Econ J 61: 830–845. https://doi.org/10.2307/1061001 doi: 10.2307/1061001
[15]	Jimenez MMS (2003) Efficiency and TFP growth in the Spanish regions: the role of human and public capital. Growth Change 34: 157–174. https://doi.org/10.1111/1468-2257.00212 doi: 10.1111/1468-2257.00212
[16]	Pereira AM, Sagales OR (2003) Spillover effects of public capital formation: evidence from the Spanish regions. J Urban Econ 53: 238–256. https://doi.org/10.1016/S0094-1190(02)00517-X doi: 10.1016/S0094-1190(02)00517-X
[17]	Pereira AM, Andraz JM (2006) Public investment in transportation infrastructures and regional asymmetries in Portugal. Ann Reg Sci 40: 803–817. https://doi.org/10.1007/s00168-006-0066-6 doi: 10.1007/s00168-006-0066-6
[18]	Kara MA, Taş S, Ada S (2016) The Impact of Infrastructure Expenditure Types on Regional Income in Turkey. Reg Stud 50: 1509–1519. https://doi.org/10.1080/00343404.2015.1041369 doi: 10.1080/00343404.2015.1041369
[19]	Geuna A, Guerzoni M, Nuccio M, et al. (2021) Resilience and Digital Disruption: Regional Competition in the Age of Industry 4.0., Berlin: Springer Nature.
[20]	Kanai JM, Schindler S (2019) Peri-urban promises of connectivity: Linking project-led polycentrism to the infrastructure scramble. Environ Plann A Econ Space 51: 302–322. https://doi.org/10.1177/0308518X18763370 doi: 10.1177/0308518X18763370
[21]	Tranos E (2011) The topology and the emerging urban geographies of the Internetbackbone and aviation networks in Europe: A comparative study. Environ Plann A Econ Space 43: 378–392. https://doi.org/10.1068/a43288 doi: 10.1068/a43288
[22]	He S, Yu S, Wang L (2021) The nexus of transport infrastructure and economic output in city-level China: a heterogeneous panel causality analysis. Ann Reg Sci 66: 113–135. https://doi.org/10.1007/s00168-020-01012-3 doi: 10.1007/s00168-020-01012-3
[23]	Lorenzetti M, Matteucci N (2016) Sviluppo socio-economico e dotazione di banda larga nei comuni marchigiani. PRISMA Economia-Società-Lavoro 3: 199–217. https://doi.org/10.3280/PRI2016-003014 doi: 10.3280/PRI2016-003014
[24]	Medeiros E (2012) Territorial Cohesion: a conceptual analysis, Regional Studies Association European Conference. 13–16.
[25]	Rauhut D, Ludlow D (2013) Services of General Interest andTerritorial Cohesion: What, How and by Whom. ESPON SeGI. Indicators and perspectives for services of general interest in territorial cohesion and development. (Draft) Scientific Report, 29–41. Available from: http://www.espon.eu.
[26]	Colomb C, Santinha G (2014) European Union Competition Policyand the European Territorial Cohesion Agenda: An impossible Reconciliation? State AidRules and Public Service Liberalization through the European Spatial Planning Lens. Eur Plan Stud 22: 459–480. https://doi.org/10.1080/09654313.2012.744384 doi: 10.1080/09654313.2012.744384
[27]	Molle W (2007) European Cohesion Policy, Abingdon: Routledge.
[28]	Van De Walle S (2009) When is a service an essential public service? Ann Public Coop Econ 80: 521–545. https://doi.org/10.1111/j.1467-8292.2009.00397.x doi: 10.1111/j.1467-8292.2009.00397.x
[29]	Suriñach J, Moreno R (2012) Introduction: Intangible Assets and Regional Economic Growth. Reg Stud 46: 1277–1281. https://doi.org/10.1080/00343404.2012.735087 doi: 10.1080/00343404.2012.735087
[30]	Gilbert MR, Masucci M (2020) Defining the Geographic and Policy Dynamics of the Digital Divide. Handbook Changing World Language Map, 3653–3671. https://doi.org/10.1007/978-3-030-02438-3_39 doi: 10.1007/978-3-030-02438-3_39
[31]	Haefner L, Sternberg R (2020) Spatial implications of digitization: State of the field and research agenda. Geogr Compass 14: e12544. https://doi.org/10.1111/gec3.12544 doi: 10.1111/gec3.12544
[32]	Huddleston JR, Pangotra PP (1990) Regional and local economic impacts of transportation investment. Transp Quartely 44: 579–594. Available from: https://hdl.handle.net/2027/uc1.c104685899?urlappend = %3Bseq = 603
[33]	Banister D, Berechman J (1999) Transport investment and economic development, 1 Eds., London: Routledge. https://doi.org/10.4324/9780203220870
[34]	Gillespie A, Robins K (1989) Geographical Inequalities: the spatial bias of the new communications technology. J Commun 39: 7–18. doi: 10.1111/j.1460-2466.1989.tb01037.x
[35]	Gibbs D, Tanner K (1997) Information and Communication Technologies and Local Economic Development Policies: The British Case. Reg Stud 31: 765–774. https://doi.org/10.1080/713693400 doi: 10.1080/713693400
[36]	Graham S (1999) Global grids of glass: on global cities, telecommunications and planetary urban networks. Urban Studies 36: 929–949. https://doi.org/10.1080/0042098993286 doi: 10.1080/0042098993286
[37]	Hackler D (2003) Invisible Infrastructure and the City: The Role of Telecommunications in Economic Development. Am Behav Sci 46: 1034–1055. https://doi.org/10.1177/0002764202250493 doi: 10.1177/0002764202250493
[38]	Baldwin R (2016) The World Trade Organization and the Future of Multilateralism. J Econ Perspect 30: 95–116. https://doi.org/10.1257/jep.30.1.95 doi: 10.1257/jep.30.1.95
[39]	Zalta EN, Nodelman U, Allen C, et al. (2005) Stanford encyclopedia of philosophy. California, CA: Stanford University Press.
[40]	Feenberg A (2017) Technosystem: the social life of reason. Cambridge, MA: Harvard University Press.
[41]	Oberhauser AM (2011) Information and Communication Technology Geographies: Strategies for Bridging a Digital Divide. Melissa R. Gilbert and Michele Masucci. Urban Geogr 32: 920–922. https://doi.org/10.2747/0272-3638.32.6.920 doi: 10.2747/0272-3638.32.6.920
[42]	Gerli P, Pontarollo E (2013) La concorrenza negata. I comportamenti strategici nella telefonia fissa, Milano: Vita e Pensiero.
[43]	Matteucci N (2019) The EU State aid policy for broadband: An evaluation of the Italian experience with first generation networks. Telecommun Policy 43: 101830. https://doi.org/10.1016/j.telpol.2019.101830 doi: 10.1016/j.telpol.2019.101830
[44]	Gòmez-Barroso JL, Perez-Martınez J (2005) Public intervention in the access to advanced telecommunication services: Assessing its theoretical economic basis. Gov Inf Q 22: 489–504. https://doi.org/10.1016/j.giq.2005.08.001 doi: 10.1016/j.giq.2005.08.001
[45]	Bonet Rull L (2021) Towards a Just Digital Transition: Urban Digital Policy in Europe After Covid-19, In: Boni AL, Zevi AT, Next Generation EU Cities: Local Communities in a post-pandemic Future, Milano: ISPI, 97–113.
[46]	National Recovery and Resilience Plan, 2021. Available from: https://www.mef.gov.it/focus/Il-Piano-Nazionale-di-Ripresa-e-Resilienza-PNRR/.
[47]	Infratel Italia, Mappatura reti fisse 2021. Available from: https://www.infratelitalia.it/archivio-documenti/documenti/esiti-mappatura-reti-fisse-2021.
[48]	Lareose R, Strover S, Gregg J, et al. (2011) The Impact of Rural Broadband Development: Lessons from a Natural Field Experiment. Gov Inf Q 28: 91–100. https://doi.org/10.1016/j.giq.2009.12.013 doi: 10.1016/j.giq.2009.12.013
[49]	Whitacre B, Gallardo R, Strover S (2014) Broadbands contribution to economic growth in rural areas: Moving towards a causal relationship. Telecommun Policy 38: 1011–1023. https://doi.org/10.1016/j.telpol.2014.05.005 doi: 10.1016/j.telpol.2014.05.005
[50]	Nevado-Peña D, López-Ruizb VR, Alfaro-Navarro JL (2019) Improving quality of life perception with ICT use and technological capacity in Europe. Technol Forecast Soc 148. https://doi.org/10.1016/j.techfore.2019.119734 doi: 10.1016/j.techfore.2019.119734
[51]	Reddick CG, Enriquez R, Harris RJ, et al. (2020) An analysis of a community survey on the digital divide. Cities 106:102904. https://doi.org/10.1016/j.cities.2020.102904 doi: 10.1016/j.cities.2020.102904
[52]	Centro Studi Camere di Commercio, Istituto Tagliacarne, Il 2018 dell'economia della provincia di Caserta e delle sue sottoaree. Disponibile su, 2018. Available from: https://www.ce.camcom.it/sites/default/files/contenuto_redazione/allegati/2018_-_nota_economica.pdf.
[53]	Galperin H, Le TV, Wyatt K (2021) Who gets access to fast broadband? Evidence from Los Angeles County. Gov Inf Q 38: 101594. https://doi.org/10.1016/j.giq.2021.101594 doi: 10.1016/j.giq.2021.101594
[54]	Infratel Italia, Relazione sullo stato di avanzamento del Progetto Nazionale Banda Ultralarga al 31 dicembre 2022. Available from: https://www.infratelitalia.it/-/media/infratel/documents/2023/relazione-stato-avanzamento-bul_dicembre2022.pdf?la = it-it & hash = 3F922319467798FF3E8DB30F230DFD4323D47B57.
[55]	Mack E, Faggian A (2013) Productivity and Broadband: The Human Factor. Int Regional Sci Rev 36: 392–423. https://doi.org/10.1177/0160017612471191 doi: 10.1177/0160017612471191
[56]	McCoy D, Lyons S, Morgenroth E, et al. (2018) The impact of broadband and other infrastructure on the location of new business establishments. J Reg Sci 58: 509–534. https://doi.org/10.1111/jors.12376 doi: 10.1111/jors.12376
[57]	World Bank (2009) World Development Report 2009: Reshaping Economic Geography, World Bank, Washington, DC.
[58]	Pike A, Rodríguez-Pose A, Tomaney J (2011) Handbook of Local and Regional Development, 1 Eds., Routledge. https://doi.org/10.4324/9780203842393
[59]	Feenberg A (2008) Critical Theory of Technology: An Overview. Tailoring Biotechnol 1: 47–64.

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Digital infrastructure strategies: the case of the province of Caserta

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Abstract

1. Introduction

2. Basic idea of the modified global error minimization method (MGEMM)

3. Examples of two nonlinear Duffing-harmonic oscillators

3.1. Example 1

3.1.1. Case 1

3.1.2. Case 2

3.2. Example 2

3.2.1. Case 1

3.2.2. Case 2

4. Results and discussion

5. Conclusions

Acknowledgments

Conflict of interest

References

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AIMS Geosciences

Digital infrastructure strategies: the case of the province of Caserta

Related Papers:

Abstract

1. Introduction

2. Basic idea of the modified global error minimization method (MGEMM)

3. Examples of two nonlinear Duffing-harmonic oscillators

3.1. Example 1

3.1.1. Case 1

3.1.2. Case 2

3.2. Example 2

3.2.1. Case 1

3.2.2. Case 2

4. Results and discussion

5. Conclusions

Acknowledgments

Conflict of interest

References

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