# Exact Solutions of Three Types of Conformable Fractional-Order Partial Differential Equations

Authors: Lijun Ma
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

Fractional calculus is widely used in biology, control systems, and engineering, so it has been highly valued by scientists. Fractional differential equations are considered an important mathematical model that is widely used in science and technology to describe physical phenomena more accurately in terms of time memory and spatial interactions. The study of exact solutions of fractional differential equations helps to understand these complex physical phenomena and dynamic processes. The results show that the method is simple and clear. In addition, with the help of MAPL, we provide some 3D maps with accurate solutions.


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## Body

## 1. Introduction

The mathematical models of many problems in life can eventually be transformed into solving integer differential equations. The development of integer integral is relatively complete from both theoretical analysis and numerical solution [1]. Fractional calculus is widely used in biology, control system, and engineering, so it has been highly valued by scientists [2]. Fractional derivatives, mainly including Caputo derivative, Riemann–Liouville derivative, and Hadamard derivative, are determined by singular integral and calculated by more complex methods [3]. The properties of conformal fractional derivative are studied. It is found that it has many properties of classical integral. Some of them were selected using MAPL and their 3D images are shown. Furthermore, we observe that this simple and easy method is an effective method for solving the fractional co-movable differential equation.Through extensive literature search, we found that fractional-order differential equations in Banach space have received a lot of attention from many scholars, especially the review of finite dimensional fractional-order differential equations and their contained edge value problems by famous foreign scholars [6], which has guided the direction for the in-depth study. Liu et al. [7] discussed the following conformable fractional-order differential equation side value problems:(1)Tαyt+ft,yt=0,y0=0,y1=λ∫01ytdt.t∈J≔0,1,Tα is a continuous function of “conformable fractional-order derivatives”, 1<α<2<f:J×R+⟶R+.Inspired by the above paper, this paper focuses on the Banach space with conformable fractional-order edge value problem (BVP):(2)Tαut=ft,ut,u0=∫0Tgs,usds,uT=∫0Ths,usds.t∈J≔0,T,The existence of the solution is unique.Tα is the “conformable fractional-order derivative,” a continuous function of 1<α<2<f,g,h:J×X⟶X, and X is the Banach space.


## 2. Related Work

Nonlinear differential equations are widely used in the modeling of complex nonlinear physical phenomena in many fields, such as electronic networks, meteorology, signal processing, and engineering science [8]. Due to the complexity of practical physical problems, when the order of derivatives in nonlinear partial differential equations is extended from integers to fractions [9], the nonlinear fractional-order partial differential equations are derived, which can more accurately characterize materials and nonlinear physical processes with time-memory properties and genetic properties [10].To address these issues, the paper presents in detail the definition of the conformable fractional-order calculus, the integral, and the linearity properties under this definition, Rolle’s theorem, and the mean value theorem [11]. A series of computational properties on the formable fractional-order calculus are given in [12], which include superposition of differential integrals, chain gauge, Gronwall’s inequality, Taylor expansion, Laplace transform, and so on. The Taylor expansion of the conformable fractional-order calculus and its fractional-order Hayashi–Steffensen inequality, Hermite–Hadamard inequality, Chebyshev inequality, Ostrowski inequality, Jensen inequality, and 11 inequalities were proved in [13]. In addition, there are many studies on the properties of fractional-order calculus, which fully illustrate that the formable fractional-order calculus has very good properties [14, 15].The method of partial integration is an important and fundamental class of methods for calculating integrals in calculus. It is derived from the multiplication rule of differentiation and the fundamental theorem of calculus. The common divisional integrals are organized in the order of divisional integrals according to the types of basic functions that make up the product function as a mnemonic: “against the power refers to three.” They refer to five types of basic functions: inverse trigonometric functions, logarithmic functions, power functions, exponential functions, and integrals of trigonometric functions, respectively.In the last six years, research on conformable fractional-order calculus has made great progress: Shrauner in [16] explored two classes of conformable linear differential equations:(3)Tαayt+1aαyt=gt,t∈a,b,a>0,ya=y0,Tαayt+ptyt=gt,t∈a,b,a>0,ya=y0.Choi et al. [17] proved the sequential linear conformable fractional-order differential equation:(4)Tnαy+pn−1tTn−1αy+⋯+p2tT2αy+p1tTαy+p0ty=0,Tnαy+pn−1tTn−1αy+⋯+p2tT2αy+p1tTαy+p0ty=qt.(5)Tnαy+pn−1Tn−1αy+⋯+p2T2αy+p1Tαy+p0y=0,Tnαy+pn−1Tn−1αy+⋯+p2T2αy+p1Tαy+p0y=qt,The above findings were all obtained for the case 0 < α ≤ 1. When α > 1, corresponding studies and analyses have also been done.Akinyemi et al. [18] extended the above conclusions and studied the existence of positive solutions to the following conformable fractional-order edge value problem:(6)Tαyt+ft,yt=0,t∈0,1,y0=0,y1=λ∫0ηytdt,where α∈1,2,η∈0,1,f:0,1×0,∞⟶0,∞, and λ is the normal number.


## 3. Preliminary

The basic concepts and lemmas used in the article are as follows.


### 3.1. Conformable Fractional-Order Calculus

Definition 1.
Letα∈n,n+1, such that β=α−n; for a given function f,g:α,+∞⟶ℜ, if fn exists, then the conformable left derivative of f is defined as(7)Tαaft=Tβafnt.
Similarly, the conformable fractional-order right derivative off is defined as(8)Tαbft=−1n+1Tβbafnt.Definition 2.
Letα∈0,1; then, the conformable fractional-order left integral of f is defined as(9)Iαaft=∫atfxdαx,a=∫atx−aα−1fxdx.
Similarly, the conformable fractional-order right integral off is defined as(10)Iαaft=∫atfxdαx,a=∫atx−aα−1fxdx,where dαb,x=b−xα−1dx.


### 3.2. Properties of the Conformable Fractional-Order Calculus

Property 1.
Iffa=fb:a,b⟶ℜ.c∈a,b is differentiable at point t0 > 0 of order a, then f is continuous at point t0.Theorem 1.
(conformable fractional-order differential median theorem). Letα > 0, and f:a,b⟶ℜ.∀t>a be satisfied:(1)
f is continuous on a,b.(2)
f is a-order differentiable, where α ∈ (0, 1).
Then, there existsc∈a,b such that Tαafc=fb−fa/1/αbα−1/αaα.Theorem 2.
Letf:a,+∞⟶ℜα∈n,n+1 be continuous and 0 < α ≤ 1; then, ∀t>a, TαaIαaft=ft.Definition 3.
LetE1 and E2 be two Banach spaces, mLPJ,R=<∞:=supu:t∈JLPJ,R. Let the operator A:D⟶E2; if A maps any bounded set S in D to a column-tight set AS in E2 (i.e., AS is relatively tight, i.e., its closure AS is a tight set in E2), then A is said to be a tight operator that maps D into E2.Definition 4.
LetE1 and E2 be two Banach spaces, D⊆E1. Let the operator A:D⟶E2. If the operator A is continuous and tight, then A is a fully continuous operator that maps D into E2.


## 3.1. Conformable Fractional-Order Calculus

Definition 1.
Letα∈n,n+1, such that β=α−n; for a given function f,g:α,+∞⟶ℜ, if fn exists, then the conformable left derivative of f is defined as(7)Tαaft=Tβafnt.
Similarly, the conformable fractional-order right derivative off is defined as(8)Tαbft=−1n+1Tβbafnt.Definition 2.
Letα∈0,1; then, the conformable fractional-order left integral of f is defined as(9)Iαaft=∫atfxdαx,a=∫atx−aα−1fxdx.
Similarly, the conformable fractional-order right integral off is defined as(10)Iαaft=∫atfxdαx,a=∫atx−aα−1fxdx,where dαb,x=b−xα−1dx.


## 3.2. Properties of the Conformable Fractional-Order Calculus

Property 1.
Iffa=fb:a,b⟶ℜ.c∈a,b is differentiable at point t0 > 0 of order a, then f is continuous at point t0.Theorem 1.
(conformable fractional-order differential median theorem). Letα > 0, and f:a,b⟶ℜ.∀t>a be satisfied:(1)
f is continuous on a,b.(2)
f is a-order differentiable, where α ∈ (0, 1).
Then, there existsc∈a,b such that Tαafc=fb−fa/1/αbα−1/αaα.Theorem 2.
Letf:a,+∞⟶ℜα∈n,n+1 be continuous and 0 < α ≤ 1; then, ∀t>a, TαaIαaft=ft.Definition 3.
LetE1 and E2 be two Banach spaces, mLPJ,R=<∞:=supu:t∈JLPJ,R. Let the operator A:D⟶E2; if A maps any bounded set S in D to a column-tight set AS in E2 (i.e., AS is relatively tight, i.e., its closure AS is a tight set in E2), then A is said to be a tight operator that maps D into E2.Definition 4.
LetE1 and E2 be two Banach spaces, D⊆E1. Let the operator A:D⟶E2. If the operator A is continuous and tight, then A is a fully continuous operator that maps D into E2.


## 4. Solution Method

### 4.1. Invariant Subspace Method

Consider the subspaceW3=L1,x,x2 , whereupon an exact solution of the following form is obtained:(11)ux,t=C1t+C2tx+C3tx2,where C1t,C2t,C3t are the functions to be determined. Substituting (11), let all coefficients of the same power of x be equal to zero:(12)dαC1tdtα=C22t+2C1tC3t,(13)dαC2tdtα=6C2tC3t,(14)dαC3tdtα=6C32t.Solving the system of equation (13) yields(15)C1t=3a22Γ1−2αΓ1−αt−α,C2t=at−α,C3t=16Γ1−αΓ1−2αt−α.An exact solution of (11) can be obtained as(16)ux,t=3a22Γ1−αΓ1−αt−α+axt−α+16Γ1−αΓ1−αx2t−α,where α is an arbitrary constant and α∈0,1/2∪1/2,1.


### 4.2. Combination of Variable Separation and Chi-Square Method

Suppose that (11) has a solution of the following form:(17)ux,t=∑k=0maktγkxk.By substituting (16) into the time fractional-order differential (11), it is easy to know that the highest number of x in ∂αu/∂tα is m, and the highest number of x in the nonlinear terms ∂u/∂x2,u∂2u/∂x2 is 2 m–2. Using the principle of chi-squared equilibrium, let m  = 2 m−2, and we can get m  = 2, so the exact solution of (11) has the following form:(18)ux,t=a0tγ0+a1tγ1x+a2tγ2x2,where γ0,γ1,γ2,a0,a1,a2 are constants to be determined. Substitute (17) into (11) to obtain(19)a0Γγ0+1Γγ0+1−αtγ0−α+a1Γγ1+1Γγ1+1−αtγ1−αx+a2Γγ2+1Γγ2+1−αtγ2−αx2=a12t2γ1+2a0a2tγ0+γ2+6a1a2tγ1+γ2x+6a22t2γ2x2.In (18), let all powers of t be equal:(20)γ0−α=γ1−α=γ2−α=2γ1=γ0+γ2=γ1+γ2=2γ2.Solving (19) yields(21)γ0=γ1=γ2=−α.In (17), using the principle of chi-squared equilibrium, let(22)a0Γγ0+1Γγ0+1−α=a12+2a0a2,a1Γγ1+1Γγ1+1−α=6a1a2,a2Γγ2+1Γγ2+1−α=6a22.Solving (22) yields(23)a0=3Γ1−2α2Γ1−αa12,a1=a1,a2=Γ1−α6Γ1−α.Substituting (21) and (23) into (17), an exact solution of (11) can be obtained as(24)ux,t=3Γ1−2α2Γ1−αa12t−α+a1t−αx+Γ1−α6Γ1−2αt−αx2,where a1 is an arbitrary constant.


### 4.3. Combination of the Flush Equilibrium and Integral Branching Methods

(1) Assimilation refers to the transformation of acquired information so that it conforms to existing cognitive styles, although this transformation may distort the information to some extent. (2) Conformity refers to the alteration and adaptation of the conceptual model of the old knowledge to accommodate the new content after the child has acquired the new knowledge. (3) Balancing refers to a dynamic process of knowledge recognition that aims at a better state of equilibrium and generally occurs when the old and new information are evenly matched and one side cannot completely consume the other [19–22].Assume that (11) has exact solutions of the following separated variable types.(25)ux,t=a0+a1vxtγ.Substituting (24) into (11) yields(26)Γ1+γΓ1+γ−αa0+a1vtγγ−α=a12dvdx2+a0a1+a12vd2vdx2t2γ.By the principle of chi-square equilibrium, let all the powers oft in (25) be equal:(27)γ−α=2γ.The solution isγ=−α, which can be obtained by substituting (26) and eliminating t−2α.(28)a0a1+a12vd2vdx2+a12dvdx2−a1Ω0v−a0Ω0=0,where Ω0=Γ1−α/Γ1−2α, such that dv/dx=y. (34) can be reduced to the following singular planar dynamical system:(29)dvdx=y,dydx=a0Ω0/a1+Ω0v−a1y2a0+a1v.It is easy to find that whenv=−a0/a1, dy/dx is meaningless and systems (28) and (29) are not equivalent. System (29) is not equivalent to (28), and v=−a0/a1 is a nontrivial solution of (28). In order to obtain a system that is completely equivalent to (28) no matter how the function v varies, the following transformation is required:(30)dx=a0+a1vdτ,where τ is a parameter, so that system (29) can be reduced to a regular planar system:(31)dx=dvdτ=a0+a1vy,dydτ=a0Ω0a1+Ω0v−a1y2.Obviously, systems (29) and (31) have the same first integral:(32)y2=2Ω0/3a12a0+a1v3+ha0+a1v2,where h is a constant of integration. (32) can be rewritten as(33)Hv,y≡a0+a1v2y2−2Ω03a12a0+a1v3=h.Obviously, system (32) has only one equilibrium point P−a0/a1,0 on the v-axis and H−a0/a1,0=0. In fact, when h=H−a0/a1,0=0, it is easy to obtain the solution vx of (31).


### 4.4. The Separation of Function Variable Method and the Separation of Generalized Variables by the Airy Equation Method

Case 1.
Application of the method of separation of functional variables [23, 24].
Suppose the solution of (34) has the following form:(34)u=Fz,z=mx+ψt,where Fz,ψt are the unknown functions to be determined. Plugging (35) into (34) yields the following generalized differential equation:(35)t1−αF′ψ′+m3F″=0.
For some constantsμ, we have(36)F′=m3μF″,and(37)−μt1−αψ′=1.
Solving (36) and (37), we have the following.(1)
Type 1: whenμ > 0, then(38)Fz=C1+C2e1/m3μz+C3e−1/m3μz.(2)
Type 2: whenμ < 0, then(39)Fz=C1+C2cos1−m3μz+C3sin1−m3μz,and(40)ψt=−tαμα+C4,where Cii=1,…,4 is an arbitrary constant and is a nonzero constant. Combining equations (38), (39), and (40), the following two traveling wave solutions are obtained:(41)u1x,t=C1+C2exp1m3μmx−tαμα+C1+C3exp−1m3μmx−tαμα+C4,and(42)u2x,t=C1+C2cos1−m3μmx−tαμα+C4+C3sin1−m3μmx−tαμα+C4.
With the help of Maple software, if we takeC1=C4=1,C2=2,C3=3,μ=−1,α=0.7 in (42), we can derive the 3D figure of u2x,t, as shown in Figure 1. (2 and 3).Figure 1
3D diagram ofu2x,t.Figure 2
3D diagram ofu3x,t and u4x,t.Figure 3
3D diagram ofu5x,t.Case 2.
Application of the generalized variable separation method.(1)
Assumptions:(43)φx=−μ6x3+C1x2+C2x+C3,ψt=μtαα+C4.(2)
Substituting (43) into (34), we get(44)φxt1−αψ′t+φ″xψt=0.(3)
For some constantsμ, we have(45)u5x,t=−μ6x3+C1x2+C2x+μtαα+C3+C4,and(46)φ″x+μφx=0.(4)
Solving (45) and (46), we have the following.(1)
Type 1: whenμ > 0, then(47)φx=C1e−μ3x+eμ3x2C2cos3μ3x2+C3sin3μ3x2.(2)
Type 2: whenμ < 0, then(48)φx=C1e−μ3x+eμ3x2−C2sin3−μ3x2+C3cos3−μ3x2,and(49)ψt=C4eμtαα/α,where Cii=1,…,4 is an arbitrary constant and C4 is a nonzero constant. In conclusion, combining equations (47)–(49) and (45), the exact solution is obtained. (34) has the following form:(50)u3x,t=C4eμtα/αC1e−μ3x+eμ3x2C2cos3μ3x2+C3sin3μ3x2,and(51)u4x,t=C4eμtα/αC1e−μ3x+eμ3x2−C2sin3μ3x2+C3cos3μ3x2.(5)
With the help of Maple software, if we takeC1=1,C2=2,C3=C4=0,μ=5,α=0.9 in (50), we can derive the 3D figure of u3x,t. If we take C1=C4=1,C2=2,C3=3,μ=−1,α=0.7, we can obtain the 3D figure of u4x,t, as shown in Figure 2.(6)
Assumptions:(52)ux,t=φx+ψt,φx≠0,ψt≠0.(7)
Substituting (52) into (34), we get(53)t1−α−μψ′t+φ″x=0.(8)
For some constantsμ, we have(54)t1−α−μψ′t=μ,and(55)−φ″x=μ.(9)
Solving equations (54) and (55), we obtain(56)φx=−μ6x3+C1x2+C2x+C3,ψt=μtαα+C4,where Cii=1,…,4 are arbitrary constants. With the help of (56) and (53), we obtain the exact solution of (34) in the following form:(57)u5x,t=−μ6x3+C1x2+C2x+μtαα+C3+C4.(10)
With the help of Maple software, if we takeC1=1,C2=2,C3=C4=0,μ=5,α=0.9 in (57), we can derive the 3D figure of u5x,t as shown in Figure 3.


## 4.1. Invariant Subspace Method

Consider the subspaceW3=L1,x,x2 , whereupon an exact solution of the following form is obtained:(11)ux,t=C1t+C2tx+C3tx2,where C1t,C2t,C3t are the functions to be determined. Substituting (11), let all coefficients of the same power of x be equal to zero:(12)dαC1tdtα=C22t+2C1tC3t,(13)dαC2tdtα=6C2tC3t,(14)dαC3tdtα=6C32t.Solving the system of equation (13) yields(15)C1t=3a22Γ1−2αΓ1−αt−α,C2t=at−α,C3t=16Γ1−αΓ1−2αt−α.An exact solution of (11) can be obtained as(16)ux,t=3a22Γ1−αΓ1−αt−α+axt−α+16Γ1−αΓ1−αx2t−α,where α is an arbitrary constant and α∈0,1/2∪1/2,1.


## 4.2. Combination of Variable Separation and Chi-Square Method

Suppose that (11) has a solution of the following form:(17)ux,t=∑k=0maktγkxk.By substituting (16) into the time fractional-order differential (11), it is easy to know that the highest number of x in ∂αu/∂tα is m, and the highest number of x in the nonlinear terms ∂u/∂x2,u∂2u/∂x2 is 2 m–2. Using the principle of chi-squared equilibrium, let m  = 2 m−2, and we can get m  = 2, so the exact solution of (11) has the following form:(18)ux,t=a0tγ0+a1tγ1x+a2tγ2x2,where γ0,γ1,γ2,a0,a1,a2 are constants to be determined. Substitute (17) into (11) to obtain(19)a0Γγ0+1Γγ0+1−αtγ0−α+a1Γγ1+1Γγ1+1−αtγ1−αx+a2Γγ2+1Γγ2+1−αtγ2−αx2=a12t2γ1+2a0a2tγ0+γ2+6a1a2tγ1+γ2x+6a22t2γ2x2.In (18), let all powers of t be equal:(20)γ0−α=γ1−α=γ2−α=2γ1=γ0+γ2=γ1+γ2=2γ2.Solving (19) yields(21)γ0=γ1=γ2=−α.In (17), using the principle of chi-squared equilibrium, let(22)a0Γγ0+1Γγ0+1−α=a12+2a0a2,a1Γγ1+1Γγ1+1−α=6a1a2,a2Γγ2+1Γγ2+1−α=6a22.Solving (22) yields(23)a0=3Γ1−2α2Γ1−αa12,a1=a1,a2=Γ1−α6Γ1−α.Substituting (21) and (23) into (17), an exact solution of (11) can be obtained as(24)ux,t=3Γ1−2α2Γ1−αa12t−α+a1t−αx+Γ1−α6Γ1−2αt−αx2,where a1 is an arbitrary constant.


## 4.3. Combination of the Flush Equilibrium and Integral Branching Methods

(1) Assimilation refers to the transformation of acquired information so that it conforms to existing cognitive styles, although this transformation may distort the information to some extent. (2) Conformity refers to the alteration and adaptation of the conceptual model of the old knowledge to accommodate the new content after the child has acquired the new knowledge. (3) Balancing refers to a dynamic process of knowledge recognition that aims at a better state of equilibrium and generally occurs when the old and new information are evenly matched and one side cannot completely consume the other [19–22].Assume that (11) has exact solutions of the following separated variable types.(25)ux,t=a0+a1vxtγ.Substituting (24) into (11) yields(26)Γ1+γΓ1+γ−αa0+a1vtγγ−α=a12dvdx2+a0a1+a12vd2vdx2t2γ.By the principle of chi-square equilibrium, let all the powers oft in (25) be equal:(27)γ−α=2γ.The solution isγ=−α, which can be obtained by substituting (26) and eliminating t−2α.(28)a0a1+a12vd2vdx2+a12dvdx2−a1Ω0v−a0Ω0=0,where Ω0=Γ1−α/Γ1−2α, such that dv/dx=y. (34) can be reduced to the following singular planar dynamical system:(29)dvdx=y,dydx=a0Ω0/a1+Ω0v−a1y2a0+a1v.It is easy to find that whenv=−a0/a1, dy/dx is meaningless and systems (28) and (29) are not equivalent. System (29) is not equivalent to (28), and v=−a0/a1 is a nontrivial solution of (28). In order to obtain a system that is completely equivalent to (28) no matter how the function v varies, the following transformation is required:(30)dx=a0+a1vdτ,where τ is a parameter, so that system (29) can be reduced to a regular planar system:(31)dx=dvdτ=a0+a1vy,dydτ=a0Ω0a1+Ω0v−a1y2.Obviously, systems (29) and (31) have the same first integral:(32)y2=2Ω0/3a12a0+a1v3+ha0+a1v2,where h is a constant of integration. (32) can be rewritten as(33)Hv,y≡a0+a1v2y2−2Ω03a12a0+a1v3=h.Obviously, system (32) has only one equilibrium point P−a0/a1,0 on the v-axis and H−a0/a1,0=0. In fact, when h=H−a0/a1,0=0, it is easy to obtain the solution vx of (31).


## 4.4. The Separation of Function Variable Method and the Separation of Generalized Variables by the Airy Equation Method

Case 1.
Application of the method of separation of functional variables [23, 24].
Suppose the solution of (34) has the following form:(34)u=Fz,z=mx+ψt,where Fz,ψt are the unknown functions to be determined. Plugging (35) into (34) yields the following generalized differential equation:(35)t1−αF′ψ′+m3F″=0.
For some constantsμ, we have(36)F′=m3μF″,and(37)−μt1−αψ′=1.
Solving (36) and (37), we have the following.(1)
Type 1: whenμ > 0, then(38)Fz=C1+C2e1/m3μz+C3e−1/m3μz.(2)
Type 2: whenμ < 0, then(39)Fz=C1+C2cos1−m3μz+C3sin1−m3μz,and(40)ψt=−tαμα+C4,where Cii=1,…,4 is an arbitrary constant and is a nonzero constant. Combining equations (38), (39), and (40), the following two traveling wave solutions are obtained:(41)u1x,t=C1+C2exp1m3μmx−tαμα+C1+C3exp−1m3μmx−tαμα+C4,and(42)u2x,t=C1+C2cos1−m3μmx−tαμα+C4+C3sin1−m3μmx−tαμα+C4.
With the help of Maple software, if we takeC1=C4=1,C2=2,C3=3,μ=−1,α=0.7 in (42), we can derive the 3D figure of u2x,t, as shown in Figure 1. (2 and 3).Figure 1
3D diagram ofu2x,t.Figure 2
3D diagram ofu3x,t and u4x,t.Figure 3
3D diagram ofu5x,t.Case 2.
Application of the generalized variable separation method.(1)
Assumptions:(43)φx=−μ6x3+C1x2+C2x+C3,ψt=μtαα+C4.(2)
Substituting (43) into (34), we get(44)φxt1−αψ′t+φ″xψt=0.(3)
For some constantsμ, we have(45)u5x,t=−μ6x3+C1x2+C2x+μtαα+C3+C4,and(46)φ″x+μφx=0.(4)
Solving (45) and (46), we have the following.(1)
Type 1: whenμ > 0, then(47)φx=C1e−μ3x+eμ3x2C2cos3μ3x2+C3sin3μ3x2.(2)
Type 2: whenμ < 0, then(48)φx=C1e−μ3x+eμ3x2−C2sin3−μ3x2+C3cos3−μ3x2,and(49)ψt=C4eμtαα/α,where Cii=1,…,4 is an arbitrary constant and C4 is a nonzero constant. In conclusion, combining equations (47)–(49) and (45), the exact solution is obtained. (34) has the following form:(50)u3x,t=C4eμtα/αC1e−μ3x+eμ3x2C2cos3μ3x2+C3sin3μ3x2,and(51)u4x,t=C4eμtα/αC1e−μ3x+eμ3x2−C2sin3μ3x2+C3cos3μ3x2.(5)
With the help of Maple software, if we takeC1=1,C2=2,C3=C4=0,μ=5,α=0.9 in (50), we can derive the 3D figure of u3x,t. If we take C1=C4=1,C2=2,C3=3,μ=−1,α=0.7, we can obtain the 3D figure of u4x,t, as shown in Figure 2.(6)
Assumptions:(52)ux,t=φx+ψt,φx≠0,ψt≠0.(7)
Substituting (52) into (34), we get(53)t1−α−μψ′t+φ″x=0.(8)
For some constantsμ, we have(54)t1−α−μψ′t=μ,and(55)−φ″x=μ.(9)
Solving equations (54) and (55), we obtain(56)φx=−μ6x3+C1x2+C2x+C3,ψt=μtαα+C4,where Cii=1,…,4 are arbitrary constants. With the help of (56) and (53), we obtain the exact solution of (34) in the following form:(57)u5x,t=−μ6x3+C1x2+C2x+μtαα+C3+C4.(10)
With the help of Maple software, if we takeC1=1,C2=2,C3=C4=0,μ=5,α=0.9 in (57), we can derive the 3D figure of u5x,t as shown in Figure 3.


## 5. Conclusion

The significant advantage of fractional-order mathematical models over integer-order integral is that they can better simulate various physical phenomena in science and technology and have a deep physical background. In addition, we also obtained a series of physical breakwaters, selected some of them using MAPL, and displayed their 3D images. In addition, we observe that such a simple and easy method is an effective method, which can be used to solve fractional compatible differential equations.



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*Source: 5295115-2022-07-30.xml*