Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag

1.   Introduction

Integral equations (IEs) form the common core in the foundations of several science and engineering principles. Thus, several computational approaches have been developed to approximate their solutions ^{[1,5,6,7,11,13,16,17,19,21,23,24,26,29,35]}.

The approximate solutions of IEs and the error behavior accompanying these solutions have been investigated in abundance. Brunner ^[10] and Maleknejad and Hadizadeh ^[22], for instance, employed the collocation methods and the Adomian decomposition method (ADM), respectively, to approach the numerical solution of the nonlinear Volterra-Fredholm integral equations (NV–FIEs) of the second kind. Wazwaz ^[28] demonstrated the use of the modified ADM (MADM) for mixed NV–FIEs of the second kind. El-Borai et al. ^[14] examined the existence and uniqueness of the solution for the NV-FIE of the second kind and discussed the normality and continuity of the integral operator. Aziz ^[8] investigated new algorithms for the numerical solution of nonlinear FIEs of the second kind using Haar wavelets. Abdou and Elkojok ^[2] investigated a numerical method for solving two-dimensional mixed nonlinear IEs with respect to time and position and discussed the existence of a unique solution of nonlinear quadratic IEs of the second kind. Furthermore, Abdou and Raad ^[3] demonstrated the ADM and its new modifications. In addition, numerical schemes are utilized by Xie et al. ^{[31,32,33,34]} to solve many types of nonlinear systems of fractional integro and partial differential equations. Brezinski and Redivo-Zaglia ^[9] explored the extrapolation methods for the numerical solution of nonlinear FIEs of the second kind. Katani ^[20] applied the quadrature methods to study the numerical solutions of FIEs. Ezquerro and Hernández-Verón ^[15] demonstrated an approach for obtaining the domains of the existence and uniqueness of the solution for FIEs, including the numerical solutions and assigning priori and posteriori error estimates for these approximations.

The phase lag is extremely important in real-life applications of IEs. Currently, there are single, dual, and three phases, with each phase tied to different applications ^[4,12]. In this study, we develop a new technique that combines the MADM and quadrature rules for describing the solution behavior of NV-FIE with the phase lag parameter.

Assuming an NV-FIE of the second kind,

tied with an initial condition

where $x = x\left(x_{1}, x_{2}, \ldots, x_n\right), y = y\left(y_{1}, y_{2}, \ldots, y_{n}\right), \ {\rm{and \;both}} \ \ \mu\neq0\ {\rm{and}}\ \lambda \neq0\ {\rm { are \;constans. }}$ Eq (1.1) will be discussed in the space $L_{2}[a, b] \times C[0, T], \ T < 1$, where $[a, b]$ is the domain of integration with respect to position while the time $t \in[0, T]$. Here, the Fredholm integral term is considered in the space $L_{2}[a, b]$ and the Volterra term is considered in the class $C[0, T]$. Moreover, Eq (1.1) possesses a unique solution under the following conditions:

1. The kernel of position $K(x, y)$ is continuous in $L_2[a, b]$ and satisfies $|K(x, y)|\leq A_1$, whereas the kernel of time $F(t, \tau)$ is continuous in $C[0, T]$ and satisfies $| F (t, \tau)| \leq B_1,$ $\forall t, \tau \in[0, T]$, and $0 \leq \tau \leq t \leq T < 1$.

2. The given function $f(x, t)$ is continuous in the space $L_{2}[a, b] \times C[0, T]$, and its norm is defined as $\|f\|_{L_{2}[a, b] \times C[0, T]} = \max\limits_{0 \leq t \leq T}\left|\int_{0}^{t}\left(\int_{a}^b f^{2}(x, \tau) d x\right)^{\frac{1}{2}} d \tau\right|\leq C_1$, whereas the unknown function $\phi(x, t)$ exhibits the same behavior as the given function with $\|\phi\| \leq C_2$.

3. (a) The known continuous function $G(x, t, \phi(x, t))$ satisfies the Lipschitz condition

$\left|G(x, t, \phi_2(x, t))-G(x, t, \phi_1(x, t))\right| \leq N(x, t)\left|\phi_{2}(x, t)-\phi_{1}(x, t)\right|$, where

$\|N\| = \max\limits_{0 \leq t \leq T}\left|\int_{0}^{t} \left(\int_{a}^{b} N^{2}(x, \tau) dx\right)^{\frac{1}{2}} d \tau\right|\leq D_1$.

(b) Furthermore, the function $G(x, t, \phi(x, t))$ satisfies the inequality

$\max\limits _{0 \leq t \leq T}\left|\int_{0}^{t}\left\{\int_{a}^{b}| G(x, \tau, \phi(x, \tau)))|^{2} dx\right\}^{\frac{1}{2}} d\tau\right| \leq D_2 \|\phi\|$,

where $A_1, B_1, C_1, C_2, D_1$, and $D_2$ are positive constants.

Theorem 1. (without proof) ^[14] If conditions (1), (2), and (3.a) are satisfied, then Eq (1.1) has a unique solution $\phi (x, t)$ in the space $L_2[a, b]\times C[0, T]$, $0\leq t \leq T < 1$, under the condition

A shock wave ^{[18,25,27,36]} is a strong pressure wave in an elastic medium, such as air, water, or a solid substance, produced by any phenomenon that drastically changes the pressure. Shock waves differ from sound waves in that the wavefront where compression occurs is a region of violent and sudden changes in the stress, density, and temperature. Therefore, shock waves travel faster than sound, and their speed increases as the amplitude is raised; however, the intensity of a shock wave decreases faster than that of a sound wave because some of its energy is expended to heat the medium in which it travels. Moreover, shock waves change the electrical, mechanical, and thermal properties of solids; therefore, they can be used to study the equation of the state of any material. The following NV-FIE of the second kind with a phase lag is obtained after the shock wave:

This study aims to discuss the stability of the solution for Eq (1.3), which has many physical implications in the fields of engineering, mathematical physics, and biology ^[13,19,29].

2.   Mixed integral equation

Applying Taylor's expansion formula and ignoring the second derivatives in Eq (1.3) results in

Accordingly, integrating Eq (2.1) w.r.t. time $t$ under the initial condition yields

where

By interchanging the integral in the plane $\tau z$ and the plane $z y$, Eq (2.2) can be expressed as

where

Applying conditions (1) and (2), the following statements can be generalized:

where $B_2, \ B_3$, and $C_3$ are positive constants.

2.1.   Stability of the solution for the mixed integral equation

This section discusses the stability of the solution for the nonlinear mixed IE represented in Eq (2.3).

Theorem 2. If conditions ($\acute{1}$), ($\acute{2}$), and (3.a) are satisfied, then Eq (2.3) has a unique and stable solution $\phi (x, t)$ in the space $L_2[a, b]\times C[0, T]$, $0\leq t \leq T < 1$, under the condition

Proof. Applying Picard's method, a solution for Eq (2.3) can be constructed as a sequence of functions $\{\phi_n(x, t)\}$ as $n\rightarrow \infty$; thus,

where

and the functions $u_n(x, t), n = 0, 1, 2, ...,$ are continuous functions defined as

Lemma 1. If the series $\sum \limits_{i = 0}^{n} u_i(x, t)$ is uniformly convergent, then $\phi(x, t)$ represents a solution of Eq (2.3).

Proof. We construct a sequence $\phi_m(x, t)$ based on

Employing Eq (2.8) to Eq (2.7) using the norm properties,

Subsequently, the mathematical induction and conditions (1), (3.a), ($\acute{1}$), and ($\acute{2}$) are applied to obtain

Therefore,

which implies the sequence $\phi_n(x, t)$ has a convergent solution. Thus, for $n \rightarrow \infty$, $\phi_n(x, t) = \sum\limits_{i = 0}^{n} u_i(x, t)$ represents a solution of Eq (2.3).

Lemma 2. The function $\phi(x, t)$ of the series (2.6) represents a unique solution of Eq (2.3).

Proof. Suppose there exists another continuous solution $\tilde{\phi}(x, t)$ of Eq (2.3), then

Note that under the given conditions, inequality (2.12) yields

As $\eta_1 < 1$, it is implied that $\phi(x, t) = \tilde{\phi}(x, t)$.

2.2.   Normality and continuity of an integral operator

To prove the normality and continuity of the reduced mixed IE (2.3), it will be first expressed in its integral operator form

where

and

2.2.1.   Normality of the integral operator

Form the norm properties,

Using the norm properties in $L_2[a, b], C[0, T]$ with the conditions (1), ($\acute{1}$), and (3.b), inequality (2.15) can be expressed as

to obtain

Therefore, the integral operator $W$ has a normality that leads directly, after using condition ($\acute{2}$), to the normality of the operator $\overline{W}$.

2.2.2.   Continuity of the integral operator

Assume two potential functions $\phi_1(x, t)$ and $\phi_2(x, t)$ in the space $L_2[a, b] \times C[0, T ]$. Applying the conditions,

and thus,

Inequality (2.17) leads to the continuity of the integral operator $W$. Furthermore, $W$ is a contraction operator in the space $L_2[a, b]\times C[0, T ]$. According to the Banach's fixed point theorem, $W$ contains a unique fixed point. If the normality and continuity of the integral operator is employed, then the existence and uniqueness of the reduced mixed IE (2.3) are approved.

3.   Modified Adomian decomposition method for NV-FIEs

Several numerical techniques can be applied to solve the NV-FIEs of the second kind ^{[19,28,29,30]}. However, herein we seek to develop a new approach that combines MADM and quadrature rules. Therefore, the solution of Eq (2.3) can be expressed as

and its approximate solution can be expressed as

Accordingly, the nonlinear term of Eq (2.3) can be decomposed into an infinite series of Adomian polynomials as

where the traditional formula of $A_{ n}(y, \tau)$ is

and the free term can be modified into the form

Consequently,

To obtain a more accurate solution for the definite integral when it could be extremely difficult in (3.5), any of these quadrature rules can be applied instead.

3.1.   Trapezoidal rule (TR)

Suppose the interval $[a, b]$ is divided into $M$ subintervals of equal width $h_t = \frac{b-a}{M}$. Then, using the equally spaced sample points $x_k = h_t k+x_0$, $k = 0, 1, 2, ..., M$, yields

Hence, (3.5) becomes

3.2.   Weddle's rule (WR)

If the interval $[a, b]$ is divided into $6m$ subintervals of equal width $h_w = \frac{b-a}{6 m}$, then applying the equally spaced sample points $y_k = h_w k+y_0$, $k = 0, 1, 2, ..., 6 m$, yields

So, (3.5) becomes

3.3.   Convergence analysis

Now, we shall present the sufficient condition for convergence of considered series.

Theorem 3. If their exists constants $\alpha \in(0, 1)$ and $k_{0} \in \mathbb{N}$ such that for each $k \geq k_{0}$, the following inequality

is satisfied, then the series solution (3.1) of Eq (2.3) is uniformly convergent in $\mathcal{I} = [a, b]\times[0, T]$.

Proof. Denoting $\mathbb{E} = (C[\mathcal{I}], \|\cdot\|)$ is the Banach space of all continuous functions on $\mathcal{I}$ with the norm $\|\phi(x, t)\| = \max\limits_{ \forall x, t \in \mathcal{I}}|\phi(x, t)|.$ Let, $\phi_{n}$ and $\phi_{m}$ be arbitrary partial sums with $n \geq m.$ We are going to prove that $\left\{\phi_{n}\right\}$ is a Cauchy sequence in $\mathbb{E}$, so we estimate the following norm

Now for any $n, k \in \mathbb{N}, n \geq k \geq k_{0},$ we have

Since $\alpha\in (0, 1)$, therefore it implies that $1-\alpha^{n-k} \leq 1$ and

So, $\left\|\phi_{n}-\phi_{k}\right\| \rightarrow 0$ as $k \rightarrow \infty,$ therefore it implies $\left\{\phi_{n}\right\}$ is a Cauchy sequence in $\mathbb{\mathbb{E}}$ and we can deduce that the series $\sum_{i = 0}^{\infty} u_{i}(x, t)$ is convergent.

3.4.   Error estimate

Next theorem concerns the estimation of error of the approximate solution $\phi_N(x, t)$.

Theorem 4. If assumptions of Theorem 3 are satisfied, $N \in \mathbb{N}$ and $N \geq k_0$, then we obtain the estimation of error of the approximate solution such that

Proof. Let $N \in \mathbb{N}$ and $N\geq k_0$, we get

In particular case, at $k_0 = 0$, we get

4.   Numerical results and discussion

In this section, the methods presented above will be utilized in some applications to explain the behavior of the solution error for some NV-FIEs of the second kind.

Application 1. Consider the NV-FIE of the second kind,

For this application, Table 1 presents the absolute values of error using MADM, MADM–TR, and MADM–WR for Eq (4.1) in the interval $x \in [0, 1]$, using different values of $t_i \in [0, 0.6], \ i = 0, 1, 2$ with $N = 3$. Here, the results were plotted in a group of Figures 1–3 to display the error behavior for each method. In addition, Table 2 lists the maximum error $E_{max}(t) = \max \limits_{i}|\phi(x_i, t)-\phi_N(x_i, t)|\ \ \forall\ x_i \in[0, 1]$ for some $t\in [0, 0.6]$.

Application 2. Consider the NV-FIE of the second kind,

Table 3 lists the absolute error values obtained using MADM, MADM-TR and MADM-WR for Eq (4.2) in the interval $x \in [0, \pi]$ at different values of $t_i \in [0, 0.4], \ i = 0, 1, 2$ with $N = 3$. Figures 4–6 show the graphically display the results that can be used to investigate the error behavior for each method. Moreover, Table 4 indicates the maximum error $E_{max}(t)\ \forall\ x_i \in[0, \pi]$ for some $t\in [0, 0.4]$.

Application 3. Consider the NV-FIE of the second kind,

Table 5 can be used to investigate the absolute value of the errors obtained using MADM, MADM–TR, and MADM–WR for Eq (4.3) in the interval $x \in [0, 1]$. The error behavior for each method at different values of $t_i\in[0, 0.2], \ i = 0, 1, 2$ with $N = 3$ is displayed in Figues 7–9. Furthermore, Table 6 shows the maximum error $E_{max}(t)\ \forall\ x_i \in[0, 1]$ for some $t\in [0, 0.2]$.

5.   Conclusions

In this paper, we focused on studying the solution of Eq (1.1), which can be interpreted with different implications in mathematical physics and in contact problems where it can be defined as

under the dynamic conditions

and where the following expression can be considered in the mathematical physics problems

Here, $\mu$ and $\lambda$ were constants; however, they could be complex and their physical implications may vary. Moreover, $\mu_i$ is the Poisson's ration and $E_i$ is the Young's coefficient of each material. IE (5.1) under conditions (5.2) was investigated throught the contact problem in the theory of elasticity of two rigid surfaces $G_i, \ i = 1, 2$ having two elastic materials occupying the contact domain $\left[0, 1\right]$ where the two functions $h_{i}(x) \in L_{2}[0, 1]$ represent and describe the equations of the upper and lower surfaces. The upper surface was impressed by a given variable force in time $\mathcal{N}_{1}(t), 0 \leq t \leq T < 1,$ with an eccentricity of application $e(t)$ and a given moment $\mathcal{N}_{2}(t)$ in consideration of the rigid displacements $\gamma(t)$ and $x\beta(t)$, respectively, through time $t \in[0, T]$ and position $x \in[0, 1].$ From the above discussions, the unknown function $\phi(x, t)$ represented the difference in the normal stresses between the two layers. Moreover, the kernel of position $K(x, y)$ depended on the properties of materials of the contact domain, whereas the known positive function $F(t, \tau)$ represented the characteristic function of the material resistance through time $t$ with $F(0, 0) =$ constant $\neq 0$.

Furthermore, the normality and continuity of NV-FIEs with phase lag in the space $L_2[a, b]\times C[0, T]$ were presented to investigate the uniqueness and existence of the solution using the Banach's fixed point theorem which is used in case of failure of Picard's method. Moreover, A new MADM based on quadrature rules, which is used in case the definite integral is extremely hard, was proposed to obtain the best approximate solutions of NV-FIEs with a phase lag. Illustrative plots of the method's applications were provided to prove the validity and accuracy of the proposed methods and to calculate the error for each method. Based on the results, the accuracy of MADM with quadrature formulas can be assigned in the order of MADM-Weddle's rule $>$ MADM-Trapezoidal rule. Thus, compared to other rules, MADM-Weddle's rule, having the same relative accuracy of MADM, is the best approach to approximate the solution of NV-FIEs.

Acknowledgments

We would like to thank Prof. Dr. A. A. Soliman, (Department of Mathematics, Faculty of Science, Benha University, Egypt) and the anonymous reviewers for their constructive suggestions towards upgrading the quality of the manuscript.

Conflict of interest

The authors declare that they have no competing interests.