Research article Special Issues

Stability in a Ross epidemic model with road diffusion

  • Received: 05 October 2021 Accepted: 15 November 2021 Published: 19 November 2021
  • MSC : 35B35, 35K60

  • Reaction-diffusion equations have been used to describe the dynamical behavior of epidemic models, where the spreading of infectious disease has the same speed in every direction. A natural question is how to describe the dynamical system when the spreading of infectious disease is directed diffusion. We introduce the road diffusion into a Ross epidemic model which describes the spread of infected Mosquitoes and humans. With the comparison principle the system is proved to have a unique global solution. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1 while the endemic equilibrium asymptotically stable if the basic reproduction number is greater than 1.

    Citation: Xiaomei Bao, Canrong Tian. Stability in a Ross epidemic model with road diffusion[J]. AIMS Mathematics, 2022, 7(2): 2840-2857. doi: 10.3934/math.2022157

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  • Reaction-diffusion equations have been used to describe the dynamical behavior of epidemic models, where the spreading of infectious disease has the same speed in every direction. A natural question is how to describe the dynamical system when the spreading of infectious disease is directed diffusion. We introduce the road diffusion into a Ross epidemic model which describes the spread of infected Mosquitoes and humans. With the comparison principle the system is proved to have a unique global solution. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1 while the endemic equilibrium asymptotically stable if the basic reproduction number is greater than 1.



    Mathematical equations that contain two or more independent variables are called partial differential equations. They are found in various scientific applications, such as chemistry, physics, engineering and mathematics, which is why researchers have developed many techniques to solve such equations as homotopy perturbation method, variation iteration method, Adomian decomposition method and others [1,2,3,4,5,6,7,8,9].

    Fractional calculus is a generalization of regular calculus, that calculates derivatives of functions of non-integer orders. Many definitions of fractional derivatives have been presented in the literature such as Riemann Louville, Caputo, conformable and others [10,11,12,13,14,15]. Furthermore, using each of these definitions can be viewed as a generalization of the normal calculus. For this reason, applying either of these definitions allows us to generalize our research to normal calculus, and since Caputo's fractional derivative is one of the most popular definitions, we establish our new results in terms of Caputo's fractional definition in this article. There are a number of different methods that mathematicians have used to solve ordinary and partial differential equations and systems in fractional calculus. One of these methods is the power series method, because it can solve such problems. On the other hand, there are too many methods that depend on the idea of power series, such as residual power series method [16,17,18,19,20,21,22,23,24,25,26], and some other techniques that combine the power series idea with transformations just as Laplace transform, ARA-transform, formable transform and others [27,28,29,30,31,32,33,34,35,36,37] to construct new ones. However, all these methods, introduce the solution in a convergent series form, but the difference lies in the level of difficulty during the applications.

    The direct power series method (DPSM) used in this work was first introduced in [38], transformations are unnecessary, no limits or differentiations are required, this method only focuses on finding the nth coefficients of the series expansion of the analytic solution.

    In DPSM we find the solution by computing a general term of the nth coefficients without going back to the hole power series every time we want to compute new coefficients. For this reason, DPSM helps mathematicians to find the approximate solutions better and faster. This method only takes two steps to write the general solution of some equations or systems that can be solved with other power series methods. At the end of the second step, we have a general form of the solution, expressed as an infinite series, and this makes it easier to find too many new coefficients of the series solution with computer programs. Furthermore, some different illustrative examples are presented in the fourth chapter and solved with the proposed method. We show that DPSM could be used to solve different types of problems and systems.

    This paper is organized as follows, in the next section, we introduce some preparatory explanations on fractional operators, Section 3 introduces the methodology and the basic idea of DPSM and finally we consider some different examples on fractional partial differential equations and systems.

    In this section we introduce the definition of the fractional Caputo and some theorems about power series.

    Definition 2.1. If ψ(χ,τ) is a function of two variables and n is any natural number, then Caputo fractional partial derivative of order α with respect to τ is denoted and defined as

    Dατψ(χ,τ)={1Γ(nα)τ0(τt)nα1nτnψ(χ,τ)dt,n1<α<n,nτnψ(χ,τ),n=α, (2.1)

    where nN.

    It is worth noting here that Caputo fractional partial derivative has the memory property. For more details and properties, see [39,40,41].

    Lemma 2.1. [10,11]. Let ϱ(χ) be a real valued continuous function and α,τ>0. Then the following properties of Caputo's derivative are hold:

    1) Dατϱ(χ)=0.

    2) Dατ(τβϱ(χ))=Γ(β+1)Γ(β+1α)τβαϱ(χ).

    3) Dkατ(τnαϱ(χ))={Γ(nα+1)Γ((nk)α+1)τ(nk)αϱ(χ),nk,0n<k.

    4) Dkατ(n=0ϱn(χ)(γτ)nαΓ(nα+1))=n=0γ(n+k)αϱn+k(χ)τnαΓ(nα+1).

    Definition 2.2. [22] For any α such that, r1<αr, rN a power series of the form:

    n=0ϱn(χ)(ττ0)nαΓ(nα+1)=ϱ0(χ)+ϱ1(χ)(ττ0)αΓ(α+1)+ϱ2(χ)(ττ0)2αΓ(2α+1)+, (2.2)

    is called the multiple fractional power series about τ=τ0, where τ is a variable and ϱn(χ),n=0,1, are functions of χ called the coefficients of the series.

    Theorem 2.1. [22,23] Suppose that ψ(χ,τ) has a fractional power series representation at τ=0, of the form:

    ψ(χ,τ)=n=0ϱn(χ)τnαΓ(nα+1),α>0,χI, (2.3)

    where 0τ<R and Ris the radius of convergence. If Dnατψ(χ,τ) is continuous on (0,R), then, the coefficients ϱn(χ); n=0,1,, of the power series (2.3) are given by

    ϱn(χ)=Dnατψ(χ,0). (2.4)

    For the proof, see [22].

    The convergence analysis of the presented power series are illustrated in the following theorem. We mention here, that these conditions are the required convergence conditions for DPSM [13].

    Theorem 2.2. [35,36] Consider the fractional power series representation in (2.3), then we have the following cases:

    a) If τ=0, the series representation (2.3) is convergent and the radius of convergence is R=0.

    b) If τ0, the series representation (2.3) is convergent and the radius of convergence is R=.

    c) If τ[0,R], the series representation (2.3) is convergent for some positive real number R and is divergent for τ>R, where R is the radius of convergence.

    In the following arguments, we state some properties of the fractional power series (2.3):

    1) The kth derivative of the fractional power series representation (2.3) is given by

    Dkχψ(χ,τ)=n=0ϱ(k)n(χ)τnαΓ(nα+1), (2.5)

    and the coefficients of the equation ϱ(m)n(χ); n=0,1, are given by

    ϱ(k)n(χ)=Dkχ(Dnατψ(χ,0)). (2.6)

    2) If we have {ψi(χ,τ)}mi=1 a sequence of functions of two variables, that has a power series expansion as follows:

    ψi(χ,τ)=n=0ϱin(χ)τnαΓ(nα+1), (2.7)

    then the coefficients of τnαΓ(nα+1) of mi=1ψi(χ,τ) can be written by the following summation:

    mi=1ψi(χ,τ)=j1,j2,jm;j1+j2++jm=nϱ1j1(χ)ϱ2j2(χ)ϱmjm(χ)Γ(nα+1)Γ(j1α+1)Γ(j2α+1)Γ(jmα+1)τnαΓ(nα+1). (2.8)

    In this section we present the main idea of the DPSM for solving time fractional partial differential equations (TFPDEs). We illustrate the technique of using DPSM to solve TFPDEs.

    Assume that ψ(χ,τ) can be presented in the series representation (2.3). Consider the FPDE,

    L[ψ(χ,τ)]+N[ψ(χ,τ)]=0, (3.1)

    where L and N denote linear and nonlinear operators respectively of fractional or integer orders derivatives. Using DPSM we can get the solution of some cases-that will be considered in Theorem 3.1, below-of Eq (3.1) in a series representation which means, to find the values of the coefficients ϱn(χ) of the series expansion (2.3). The Caputo fractional derivative affects the series formula and changes the summation, so we need to illustrate the following theorem that study the effects of the fractional derivatives on the power series representations.

    Theorem 3.1. [32] Suppose that ψ(χ,τ) and φ(χ,τ) have fractional power series representations such as:

    ψ(χ,τ)=n=0ϱn(χ)τnαΓ(nα+1)andφ(χ,τ)=m=0ρm(χ)τmαΓ(mα+1), (3.2)

    where ψ(χ,τ) and φ(χ,τ) are analytical functions, then we haveϱn(χ) and ρm(χ) are the coefficients of τnαΓ(nα+1)and τmαΓ(mα+1) in ψ(χ,τ) and φ(χ,τ) respectively,m=0,1,, and n=0,1,. Then we have:

    a) ϱn+k(χ) is the coefficient for τnαΓ(nα+1) in the series expansion of Dkατψ(χ,τ) for any k=0,1,.

    b) γ(n+k)αϱn+k(χ) is the coefficient for τnαΓ(nα+1) in the series expansion of Dkατψ(χ,γτ) for any k=0,1,, where γR.

    c) ni=0ϱi(χ)ρni(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1) is the coefficient for τnαΓ(nα+1) in the series expansion of ψ(χ,τ)φ(χ,τ).

    d) ni=0βiαγ(ni)αϱi(χ)ρni(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1) is the coefficient for τnαΓ(nα+1) in the series expansion of ψ(χ,βτ)φ(χ,γτ), where β and γR.

    e) ni=0β(i+k)αγ(ni+m)αϱi+k(χ)ρni+m(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1) is the coefficient for τnαΓ(nα+1) in the series expansion of Dkατψ(χ,βτ)Dsατφ(χ,γτ), where β, γRandk,s=0,1,.

    Proof. Substituting the series expansion of ψ(χ,τ) in Dkατψ(χ,τ),

    Dkατψ(χ,τ)=Dkατ(n=0ϱn(χ)τnαΓ(nα+1)). (3.3)

    Using part (4) of Lemma 2.1, we get

    Dkατψ(χ,τ)=n=jϱn(χ)τ(nk)αΓ((nk)α+1). (3.4)

    Thus, the series expansion can be written as

    Dkατψ(χ,τ)=n=0ϱn+k(χ)τnαΓ(nα+1). (3.5)

    Proof. Substituting the series expansion of ψ(χ,γτ)in Dkατψ(χ,γτ), we get

    Dkατψ(χ,γτ)=Dkατ(n=0ϱn(χ)(γτ)nαΓ(nα+1)). (3.6)

    Using part (4) of Lemma 2.1, we get

    Dkατψ(χ,γτ)=n=jγnαϱn(χ)τ(nk)αΓ((nk)α+1). (3.7)

    Thus, Eq (3.7), can be written as

    Dkατψ(χ,γτ)=n=0γ(n+k)αϱn+k(χ)τnαΓ(nα+1). (3.8)

    Proof. Multiplying the series expansion of ψ(χ,τ) and φ(χ,τ),

    ψ(χ,τ)φ(χ,τ)=n=0ϱn(χ)τnαΓ(nα+1)m=0ρm(χ)τmαΓ(mα+1). (3.9)

    Equation (3.9), can be simplified as

    ψ(χ,τ)φ(χ,τ)=n=0m=0ϱn(χ)ρm(χ)τ(n+m)αΓ(nα+1)Γ(mα+1), (3.10)

    which can be rewritten as

    ψ(χ,τ)φ(χ,τ)=n=0(ni=0ϱi(χ)ρni(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1))τnαΓ(nα+1). (3.11)

    Proof. Multiplying the series expansion of ψ(χ,βτ) and φ(χ,γτ),

    ψ(χ,βτ)φ(χ,γτ)=(n=0ϱn(χ)(βτ)nαΓ(nα+1))(m=0ρm(χ)(γτ)mαΓ(mα+1)). (3.12)

    Equation (3.12), can be written as

    ψ(χ,βτ)φ(χ,γτ)=n=0m=0ϱn(χ)ρm(χ)βnαγmατ(n+m)αΓ(nα+1)Γ(mα+1), (3.13)

    which can be simplified as

    ψ(χ,βτ)φ(χ,γτ)=n=0(ni=0βiαγ(ni)αϱi(χ)ρni(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1))τnαΓ(nα+1). (3.14)

    Proof. Substituting the series expansion of ψ(χ,γτ) and φ(χ,γτ)in Dkατψ(χ,βτ)Dsατφ(χ,γτ), we get:

    Dkατψ(χ,βτ)Dsατφ(χ,γτ),=Djαt(i=0ϱi(χ)(βτ)iαΓ(iα+1))Dmαt(j=0ρj(χ)(γτ)kαΓ(kα+1)). (3.15)

    Using part (2) of Lemma 2.1, Eq (3.15) can be written as

    Dkατψ(χ,βτ)Dsατφ(χ,γτ)=(n=0β(n+k)αϱn+k(χ)τnαΓ(nα+1))(m=0γ(m+s)αρm+s(χ)τmαΓ(mα+1)), (3.16)

    which can be simplified as

    Dkατψ(χ,βτ)Dsατφ(χ,γτ)=n=0m=0β(n+k)αγ(m+s)αϱn+k(χ)ρm+s(χ)τ(n+m)αΓ(nα+1)Γ(mα+1). (3.17)

    Equation (3.17) can be written as

    Dkατψ(χ,βτ)Dsατφ(χ,γτ)=n=0(ni=0β(i+k)αγ(ni+m)αϱi+k(χ)ρni+m(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1))τnαΓ(nα+1). (3.18)

    The proof is complete.

    The main idea of DPSM depends on replacing each part of the target equation or system with its τnαΓ(nα+1) coefficients, in any equation that contains similar terms in Theorem 3.1, these replacements can be applied separately for each additive part, or if each part is multiplied by a real number. The main idea of the method is to do some replacements in the target equation and simplify the obtained series expansions in one series after simple computations, to get a general term of the coefficients in the series expansion (2.3) and hence, we get the analytic series solution of the equation by substituting n=1,2,in the series form and so on.

    Our goal in this section, is to explain the usage of DPSM in solving some TFPDEs and get numerical solutions for them. The method is basically, depends on assuming the series representation (2.3) of the solution and then find a general term of the series coefficients, that allows researchers to get better approximate solutions by getting many terms of the numerical solution they study.

    The following steps illustrate the algorithm of DPSM in solving TFPDEs:

    Step 3.1. Apply the replacements from Theorem 3.1 that is, replace each term of the target equation by its suitable similar coefficient ϱn(χ) of τnαΓ(nα+1).

    Step 3.2. Simplify the obtained series representations from Step 3.1, and define a general form of the series solution by putting the higher index ϱn+k(χ) to the left-hand side and the rest of the coefficients into the right, to get a formula of the shape:

    ϱn+k(χ)=ρ(an+k1,an+k2,,a0). (3.19)

    Step 3.3. Substitute the values of n, recursively from n=1,2, as much as you need terms of the series solution.

    In this section, some interesting examples on TFPDEs are solved, we clarify the steps of DPSM by solving fractional partial differential equations and system of fractional partial differential equations, each example is of different kind.

    Example 4.1.[26] Consider the following temporal-fractional Burger equation of the form:

    Dατψ(χ,τ)ψχχ(χ,τ)+ψ(χ,τ)ψχ(χ,τ)=0,0<α1, (4.1)

    subject to the initial condition,

    ψ(χ,0)=2χ. (4.2)

    Note that, when α=1 the exact solution of Eq (4.1) the integer case is

    ψ(χ,τ)=2χ1+2τ. (4.3)

    Solution 4.1. Applying the replacements of Theorem 3.1 on Eq (4.1), we get

    Dατψ(χ,τ)ϱn+1(χ),
    ψχχ(χ,τ)ϱ''n(χ),

    and

    ψ(χ,τ)ψχ(χ,τ)ni=0ϱi(χ)ϱ'ni(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1).

    Substituting the new terms from the replacements into Eq (4.1), we get

    ϱn+1(χ)ϱ''n(χ)+ni=0ϱi(χ)ϱ'ni(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1)=0. (4.4)

    Then the Eq (4.1) can be expressed as

    ϱn+1(χ)=ϱ''n(χ)ni=0ϱi(χ)ϱ'ni(χ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1). (4.5)

    From the initial condition in (4.2),

    for n=0, ϱ1(χ)=4χ,

    for n=1, ϱ2(χ)=16χ,

    for n=2, ϱ3(χ)=16χ(4+Γ(2α+1)Γ2(α+1)),

    for n=3, ϱ4(χ)=64χ(4+Γ(2α+1)Γ2(α+1)+2Γ(3α+1)Γ(2α+1)Γ(α+1)),

    for n=4, ϱ5(χ)=256χ(4+Γ(2α+1)Γ2(α+1)+2Γ(3α+1)Γ(2α+1)Γ(α+1)+Γ(4α+1)Γ2(2α+1))128χ(4+Γ(2α+1)Γ2(α+1))Γ(4α+1)Γ(3α+1)Γ(α+1), which are the same results obtained in [26]. DPSM save a lot of time in calculating the coefficients and the steps here are written completely.

    Thus, the solution of Eqs (4.1) and (4.2), can be expressed as

    ψ(χ,τ)=ϱ0(χ)+ταϱ1(χ)Γ(1+α)+τ2αϱ2(χ)Γ(1+2α)+τ3αϱ3(χ)Γ(1+3α)+τ4αϱ4(χ)Γ(1+4α)+τ5αϱ5(χ)Γ(1+5α)
    =2χ4χταΓ(1+α)+16χτ2αΓ(1+2α)16χ(4+Γ(2α+1)Γ2(α+1))τ3αΓ(1+3α)
    +64χ(4+Γ(2α+1)Γ2(α+1)+2Γ(3α+1)Γ(2α+1)Γ(α+1))τ4αΓ(1+4α)
    +(256χ(4+Γ(2α+1)Γ2(α+1)+2Γ(3α+1)Γ(2α+1)Γ(α+1)+Γ(4α+1)Γ2(2α+1))128χ(4+Γ(2α+1)Γ2(α+1))Γ(4α+1)Γ(3α+1)Γ(α+1))τ5αΓ(1+5α)+. (4.6)

    Substituting α=1we get

    ψ(χ,τ)=2χ4τχ+8τ2χ16τ3χ+32τ4χ64τ5χ+128τ6χ
    =2χ(12τ+4τ28τ3+16τ432τ5χ+64τ6χ), (4.7)

    this result agrees, with the Maclaurin series of the exact solution of Eq (4.1) in the integer case, which is

    ψ(χ,τ)=2χ1+2τ. (4.8)

    The following figures illustrate some simulations of Example 4.1.

    In Figure 1, we sketch the solution of Example 4.1 in 3D space with α=1. We compare the exact solution of the integer order case of Example 4.1 with the fifth approximate solution from DPSM and sketch the error in Figure 2. Figure 3 present the contour graphs of the solution with different values of α.

    Figure 1.  The 3D surface plot of the solution ψ(χ,τ) for Example 4.1 with α=1,2χ2 and 2τ2.
    Figure 2.  The 3D surface plot of (a) exact solution ψ(χ,τ), (b) the fifth approximation solution ψ5(χ,τ) and (c) the absolute error of the exact solution and approximation solution, with α=1,0χ1 and 0τ0.2.
    Figure 3.  The contour graph of the solution ψ(χ,τ) of the fractional Burger equation at several values of α in Example 4.1.

    Example 4.2. [28] Consider the following time-fractional Phi-4 equation of the form:

    D2ατψ(χ,τ)=ψχχ(χ,τ)μ2ψ(χ,τ)λψ3(χ,τ), (4.9)

    with initial conditions,

    ψ(χ,0)=μ2λtanh(μχ12(ν21)), (4.10)
    Dατψ(χ,0)=μνμ2λ(ν21)sech2(μχ12(ν21)), (4.11)

    where ν is the speed of the traveling wave αϵ(0,1). The exact solution for α=1 is given by [28],

    ψ(χ,τ)=μ2λtanh(μ(χvτ)12(ν21)). (4.12)

    Solution 4.2. The solution by DPSM can be obtained replacing each part of Eq (4.9) with its suitable coefficient of τnαΓ(nα+1) as in Theorem 3.1 to get the following:

    ϱn+2(χ)=ϱ''n(χ)μ2ϱn(χ)λi,j,ki+j+k=nϱi(χ)ϱj(χ)ϱk(χ)Γ((i+j+k)α+1)Γ(iα+1)Γ(jα+1)Γ(kα+1), (4.13)

    from the initial conditions we have that ϱ0(χ)=ψ(χ,0) and ϱ1(χ)=Dατψ(χ,0).

    For n=0,

    ϱ2(χ)=ϱ''0(χ)μ2ϱ0(χ)λϱ30(χ)
    =λ(μ2λ)3/2ν2Sech2(μ11+ν2χ2)Tanh(μ11+ν2χ2)1+ν2. (4.14)

    For n=1,

    ϱ3(χ)=ϱ''1(χ)μ2ϱ1(χ)3λϱ1(χ)ϱ20(χ)
    =μ3ν3(2+cosh(2μ11+ν2χ))sech4(μ11+ν2χ2)μλ(1+ν2)2(1+ν2). (4.15)

    For n=2,

    ϱ4(χ)=ϱ''2(χ)μ2ϱ2(χ)3λϱ21(χ)ϱ0(χ)3λϱ2(χ)ϱ20(χ)
    =12(ν21)2λ(ν21)λμν2(μ2λ)3/2tanh(μ1ν21χ2)sech4(μ1ν21χ2)
    (2μλ(ν21)(ν2cosh(2μ1ν21χ)2ν23)3λ(ν21)2). (4.16)

    For n=3,

    ϱ5(χ)=ϱ''3(χ)μ2ϱ3(χ)λϱ31(χ)6λϱ0(χ)ϱ1(χ)ϱ2(χ)3λϱ3(χ)ϱ30(χ)
    =182(ν21)2μ3ν3(μλ(ν21))3/2[sech7(μ1ν21χ2)μλ(ν21)48ν2μ2λsinh(μ1ν21χ2)
    21ν2μ2λsinh(3μ1ν21χ2)+3ν2μ2λsinh(5μ1ν21χ2)48μ2λsinh(μ1ν21χ2)
    +21μ2λsinh(3μ1ν21χ2)3μ2λsinh(5μ1ν21χ2)+ν2(cosh(5μ1ν21χ2))
    +20(ν2+3)cosh(μ1ν21χ2)(11ν2+39)cosh(3μ1ν21χ2)
    +3cosh(5μ1ν21χ2)+4λ(ν21)2cosh(μ1ν21χ2)], (4.17)

    and then the 5th truncated series for ψ(χ,τ) can be expressed as,

    ψ5(χ,τ)=ϱ0(χ)+ϱ1(χ)(ττ0)Γ(α+1)+5i=2ϱi(χ)(tt0)iαΓ(iα+1)
    =μ2λtanh(μ1ν21χ2)μνsech2(μ1ν21χ2)μλ(ν21)(ττ0)2Γ(α+1)
    +λ(μ2λ)3/2ν2sech2(μ1ν21χ2)tanh(μ1ν21χ2)(ττ0)2α(ν21)Γ(2α+1)
    μ3ν3(cosh(2μ1ν21χ)2)sech4(μ1ν21χ2)μλ(ν21)(ττ0)3α2(ν21)Γ(3α+1)
    +λμ(μ2λ)3/2ν2sech4(μ1ν21χ2)tanh(μ1ν21χ2)(2μ(cosh(2μ1ν21χ)ν22ν23)λ(ν21)3λ(ν21)2)(ττ0)4α2(ν21)2Γ(4α+1)λ(ν21)
    +182(ν21)2Γ(5α+1)μ3ν3sech7(μ1ν21χ2)(μλ(ν21))3/2[4λcosh(μ1ν21χ2)(ν21)2
    +μ[cosh(5μ1ν21χ2)ν2+48μ2λsinh(μ1ν21χ2)ν221μ2λsinh(3μ1ν21χ2)ν2
    +3μ2λsinh(5μ1ν21χ2)ν2+20(ν2+3)cosh(μ1ν21χ2)(11ν2+39)cosh(3μ1ν21χ2)
    +3cosh(5μ1ν21χ2)48μ2λsinh(μ1ν21χ2)+21μ2λsinh(3μ1ν21χ2)
    3μ2λsinh(5μ1ν21χ2)]λ(ν21)](ττ0)5α. (4.18)

    We mention, that it's hard to find the fifth term by other analytical methods. By DPSM we can get many terms of the series solution by computer software.

    In the following Figures 4 and 5, we sketch the exact solution of Example 4.2 in the integer case and compare it with approximate solution of fifth order in 2D and 3D. Also, we sketch the application solution in the 2D plane with different values of to show how the approximation solution converges to the exact solution of integer order.

    Figure 4.  The 3D surface plot of (a) exact solution ψ(χ,τ), (b) the fifth approximation solution ψ5(χ,τ) and (c) the absolute error of the exact solution and approximation solution, for time-fractional Phi-4 at τ(0,1), χ(10,10), α=1,μ=1,λ=1,υ=3.
    Figure 5.  The 2D plot of (a) approximation solution ψ5(χ,τ) and exact solution ψ(χ,τ)for time-fractional Phi-4 at τ(0,1), χ=0.6, α=1,μ=1,λ=1,υ=3, (b) the DPSM solutions of different values of α in Example 4.2.

    Example 4.3. [22] The following form is considered:

    Dατψ(χ,τ)+φχχ(χ,τ)+2(ψ2(χ,τ)+φ2(χ,τ))φ(χ,τ)=0, (4.19)
    Dατφ(χ,τ)ψχχ(χ,τ)2(ψ2(χ,τ)+φ2(χ,τ))ψ(χ,τ)=0, (4.20)

    with initial conditions,

    ψ(χ,0)=cos(χ), (4.21)
    φ(χ,0)=sin(χ). (4.22)

    To find the general form of solutions by DPSM just do the replacements in Theorem 3.1 which could be directly written by,

    ϱn+1(χ)=ρ''n(χ)2i,j,ki+j+k=n(ϱi(χ)ϱj(χ)ρk(χ)+ρi(χ)ρj(χ)ρk(χ))Γ(nα+1)Γ(iα+1)Γ(jα+1)Γ(kα+1), (4.23)
    ρn+1(χ)=ϱ''n(χ)+2i,j,ki+j+k=n(ϱi(χ)ϱj(χ)ϱk(χ)+ρi(χ)ρj(χ)ϱk(χ))Γ(nα+1)Γ(iα+1)Γ(jα+1)Γ(kα+1). (4.24)

    For n=0,

    ϱ1(χ)=δρ''0(χ)γ(ϱ0(χ)ϱ0(χ)ρ0(χ)+ρ0(χ)ρ0(χ)ρ0(χ))φ(χ)ρ0(χ)=sinx,
    ρ1(χ)=δϱ''0(χ)+γ(ϱ0(χ)ϱ0(χ)ϱ0(χ)+ρ0(χ)ρ0(χ)ϱ0(χ))+φ(χ)ϱ0(χ)=cosx.

    For n=1,

    ϱ2(χ)=cosx,
    ρ2(χ)=sinx.

    For n=2,

    ϱ3(χ)=(52Γ(1+2α)Γ(1+α)2)sinx,
    ρ3(χ)=(52Γ(1+2α)Γ(1+α)2)cosx.

    For n=3,

    ϱ4(χ)=(52Γ(1+2α)Γ(1+α)2+4Γ(1+3α)Γ(1+α)Γ(1+2α)2Γ(1+3α)Γ(1+α)3)cosx,
    ρ4(χ)=(52Γ(1+2α)Γ(1+α)2+4Γ(1+3α)Γ(1+α)Γ(1+2α)2Γ(1+3α)Γ(1+α)3)sinx,

    which is the same general form of the solutions obtained in [22].

    Example 4.4. [37] Consider the following time-fractional 3-dimentional Navier Stokes equation:

    Dατψ1(χ,ς,ζ,τ)+ψ1(χ,ς,ζ,τ)ψ1(χ,ς,ζ,τ)χ+ψ2(χ,ς,ζ,τ)ψ1(χ,ς,ζ,τ)ς+ψ3(χ,ς,ζ,τ)ψ1(χ,ς,ζ,τ)ζ
    =υ(2ψ1(χ,ς,ζ,τ)χ2+2ψ1(χ,ς,ζ,τ)ς2+2ψ1(χ,ς,ζ,τ)ζ2), (4.25)
    Dατψ2(χ,ς,ζ,τ)+ψ1(χ,ς,ζ,τ)ψ2(χ,ς,ζ,τ)χ+ψ2(χ,ς,ζ,τ)ψ2(χ,ς,ζ,τ)ς+ψ3(χ,ς,ζ,τ)ψ2(χ,ς,ζ,τ)ζ
    =υ(2ψ2(χ,ς,ζ,τ)χ2+2ψ2(χ,ς,ζ,τ)ς2+2ψ2(χ,ς,ζ,τ)ζ2), (4.26)
    Dατψ3(χ,ς,ζ,τ)+ψ1(χ,ς,ζ,τ)ψ3(χ,ς,ζ,τ)χ+ψ2(χ,ς,ζ,τ)ψ3(χ,ς,ζ,τ)ς+ψ3(χ,ς,ζ,τ)ψ3(χ,ς,ζ,τ)ζ
    =υ(2ψ3(χ,ς,ζ,τ)χ2+2ψ3(χ,ς,ζ,τ)ς2+2ψ3(χ,ς,ζ,τ)ζ2), (4.27)

    where φR, and with the ICs,

    ψ1(χ,ς,ζ,0)=0.5χ+ς+ζ, (4.28)
    ψ2(χ,ς,ζ,0)=χ0.5ς+ζ, (4.29)
    ψ3(χ,ς,ζ,0)=χ+ς0.5ζ. (4.30)

    The exact solution, when α=1, of the integer case is

    ψ1(χ,ς,ζ,τ)=0.5χ+ς+ζ2.25χτ12.25τ2,
    ψ2(χ,ς,ζ,τ)=χ0.5ς+ζ2.25ςτ12.25τ2,
    ψ3(χ,ς,ζ,τ)=χ+ς0.5ζ2.25ζτ12.25τ2.

    Solution 4.3. The solution by DPSM can be obtained by replacing each part of Eqs (4.25)–(4.27) with its suitable coefficient of τnαΓ(nα+1) as in Theorem 3.1 to get the following relations:

    ϱn+1(χ,ς,ζ)=ni=0(ϱi(χ,ς,ζ)ϱni(χ,ς,ζ)χ+ρi(χ,ς,ζ)ϱni(χ,ς,ζ)ς+σi(χ,ς,ζ)ϱni(χ,ς,ζ)ζ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1)
    +υ(2ϱn(χ,ς,ζ)χ2+2ϱn(χ,ς,ζ)ς2+2ϱn(χ,ς,ζ)ζ2),
    ρn+1(χ,ς,ζ)=ni=0(ϱi(χ,ς,ζ)ρni(χ,ς,ζ)χ+ρiρni(χ,ς,ζ)ς+σiρni(χ,ς,ζ)ζ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1)
    +υ(2ρn(χ,ς,ζ)χ2+2ρn(χ,ς,ζ)ς2+2ρn(χ,ς,ζ)ζ2),
    σn+1(χ,ς,ζ)=ni=0(ϱi(χ,ς,ζ)σni(χ,ς,ζ)χ+σiσni(χ,ς,ζ)ς+σiσni(χ,ς,ζ)ζ)Γ(nα+1)Γ(iα+1)Γ((ni)α+1)
    +υ(2σn(χ,ς,ζ)χ2+2σn(χ,ς,ζ)ς2+2σnζ2).

    For n=0,

    ϱ1=ϱ0ϱ0χρ0ϱ0ςσ0ϱ0ζ+υ(2ϱ0χ2+2ϱ0ς2+2ϱ0ζ2)
    =(0.5χ+ς+ζ)(0.5)(χ0.5ς+ζ)(1)(χ+ς0.5ζ)(1)=2.25χ,
    ρ1=ϱ0ρ0χρ0ρ0ςσ0ρ0ζ+υ(2ρ0χ2+2ρ0ς2+2ρ0ζ2)
    =(0.5χ+ς+ζ)(1)(χ0.5ς+ζ)(0.5)(χ+ς0.5ζ)(1)=2.25ς,
    σ1=ϱ0σ0χρ0σ0ςσ0σ0ζ+υ(2σ0χ2+2σ0ς2+2σ0ζ2)
    =(0.5χ+ς+ζ)(1)(χ0.5ς+ζ)(1)(χ+ς0.5ζ)(0.5)=2.25ζ.

    For n=1,

    ϱ2=ϱ1ϱ0χϱ0ϱ1χρ1ϱ0ςρ0ϱ1ςσ1ϱ0ζσ0ϱ1ζ+υ(2ϱ1χ2+2ϱ1ς2+2ϱ1ζ2)
    =(2.25χ)(0.5)(0.5χ+ς+ζ)(2.25)(2.25ς)(1)(2.25ζ)(1)
    =2(0.5)(2.25)χ+2(2.25)ς+2(2.25)ζ=2(2.25)(0.5χ+ς+ζ)=4.5ϱ0,
    ρ2=ϱ1ρ0χϱ0ρ1χρ1ρ0ςρ0ρ1ςσ1ρ0ζσ0ρ1ζ+υ(2ρ1χ2+2ρ1ς2+2ρ1ζ2)
    =2(2.25)χ2(0.5)(2.25)ς+2(2.25)ζ=2(2.25)(χ0.5ς+ζ)=4.5ρ0,
    σ2=ϱ1σ0χϱ0σ1χρ1σ0ςρ0σ1ςσ1σ0ζσ0σ1ζ+υ(2σ1χ2+2σ1ς2+2σ1ζ2)
    =2(2.25)χ+2(2.25)ς2(0.5)(2.25)ζ=2(2.25)(χ+ς0.5ζ)=4.5σ0,

    For n=2,

    ϱ3=(20.25+5.0625Γ(2α+1)Γ2(α+1))χ,
    ρ3=(20.25+5.0625Γ(2α+1)Γ2(α+1))ς,
    σ3=(20.25+5.0625Γ(2α+1)Γ2(α+1))ζ.

    For n=3,

    ϱ4=(40.5+10.125Γ(2α+1)Γ2(α+1)+20.25Γ(3α+1)Γ(2α+1)Γ(α+1))(0.5χ+ς+ζ),
    ρ4=(40.5+10.125Γ(2α+1)Γ2(α+1)+20.25Γ(3α+1)Γ(2α+1)Γ(α+1))(χ0.5ς+ζ),
    σ4=(40.5+10.125Γ(2α+1)Γ2(α+1)+20.25Γ(3α+1)Γ(2α+1)Γ(α+1))(χ+ς0.5ζ).

    So, the solution of systems (4.25)–(4.27) and (4.28)–(4.30) has the following series form:

    ψ1(χ,ς,ζ,τ)=0.5χ+ς+ζ2.25Γ(1+α)χτα
    +2(2.25)Γ(1+2α)(0.5χ+ς+ζ)τ2α(2.25)2Γ(1+3α)(4+Γ(1+2α)Γ2(1+α))χτ3α
    +(2.25)2Γ(1+4α)(8+2Γ(1+2α)Γ2(1+α)+4Γ(1+3α)Γ(1+α)Γ(1+2α))(0.5χ+ς+ζ)τ4α+, (4.31)
    ψ2(χ,ς,ζ,τ)=χ0.5ς+ζ2.25Γ(1+α)ςτα
    +2(2.25)Γ(1+2α)(χ0.5ς+ζ)τ2α(2.25)2Γ(1+3α)(4+Γ(1+2α)Γ2(1+α))ςτ3α
    +(2.25)2Γ(1+4α)(8+2Γ(1+2α)Γ2(1+α)+4Γ(1+3α)Γ(1+α)Γ(1+2α))(χ0.5ς+ζ)τ4α+, (4.32)
    ψ3(χ,ς,ζ,τ)=χ+ς0.5ζ2.25Γ(1+α)ζτα
    +2(2.25)Γ(1+2α)(χ+ς0.5ζ)τ2α(2.25)2Γ(1+3α)(4+Γ(1+2α)Γ2(1+α))ζτ3α
    +(2.25)2Γ(1+4α)(8+2Γ(1+2α)Γ2(1+α)+4Γ(1+3α)Γ(1+α)Γ(1+2α))(χ+ς0.5ζ)τ4α+. (4.33)

    The following figure (Figure 6) illustrates the graph of the exact solution of Example 4.3 and the 10th approximation solutions.

    Figure 6.  The 3D surface plot of the 10th approximate solutions of ψ1,ψ2, and ψ3 at various values of α and τ=0.5 & ζ=3 for the problem in Example 4.3. (a) α=0.6, (b) α=0.8, (c) α=1, (d) α=1 (exact solutions).

    In several types of science like mathematics, physics, engineering statics, etc., there are massive numbers of equations and systems that need solutions. Mathematicians created and developed many analytical and numerical methods to find the solution accurately or approximately. Power series methods like residual power series, Laplace residual power series, and many others give exact solutions or sometimes provide approximate solutions that converge to the exact one. However, these methods require a lot of steps each time, especially for solving nonlinear equations and systems. In this paper, a new technique has been presented for the first time called the DPSM for solving TFPDEs. In the following, we mention some remarks on the new method

    1) DPSM introduces an analytical series solution depending on the idea of power series representation.

    2) The method is simple and applicable in presenting series solutions.

    3) DPSM can obtain many terms of the series solution.

    4) The proposed method, need no discretization, transformation differentiation or taking limit.

    5) All numerical results obtained in this article by Mathematica 3.12.

    This method is a simple method among all analytical methods. It depends on the idea of the power series expansion for solving time-fractional differential equations or systems. In the future, we attend to expand the replacements and solve integral equations.

    The authors declare no conflicts of interest.



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