# Three-Dimensional Fourth-Order Time-Fractional Parabolic Partial Differential Equations and Their Analytical Solution

Authors: Yesuf Obsie Mussa; Ademe Kebede Gizaw; Ayana Deressa Negassa
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this study, the fractional reduced differential transform method (FRDTM) is employed to solve three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients. The fractional derivative used in this study is in the Caputo sense. A few important lemmas which are essential to solve the problems using the proposed method are proved. The novelty of this method is that it uses appropriate initial conditions and finds the solution to the problems without any discretization, linearization, perturbation, or any restrictive assumptions. Two numerical examples are considered in order to validate the efficiency and reliability of the method. Furthermore, the FRDTM solution whenα = 1 is compared with other analytical methods available in the existing literature. Computational results are shown in tables and graphs. The obtained results revealed that the method is capable and simple to solve fractional partial differential equations. The software used for the calculations in this study is Mathematica 7.


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## Body

## 1. Introduction

In the last decade, fractional calculus has become an important mathematical tool in several branches of sciences and engineering due to its enormous numbers of applications ([1] and the references therein). What specifics fractional calculus is the existence of various approaches to define differentiation with the nonintegral order [2]. As a result, several definitions for fractional derivatives appear in the literature to present more accurate models for real life phenomena. Some of known fractional derivatives are Riemann–Liouville, modified Riemann–Liouville, Caputo, Hadmard, Erdélyi–Kober, Riesz, Grunwald–Letnikov, Marchaud, and others ([3] and the references therein). All known fractional derivatives satisfy one of the well-known properties of classical derivative, namely, the linear property. However, the other properties of classical derivative, such as the derivative of a constant is zero; the product rule, quotient rule, and the chain rule either do not hold or are too complicated for many fractional derivatives.In recent years, fractional differential equations have received considerable attention owing to their applicability in different fields of sciences such as chemistry, biology, diffusion, control theory, rheology, viscoelasticity, and so on [4]. Consequently, the solution of FPDEs represents nowadays a vigorous research area for scientists, and finding approximate and exact solutions to FPDEs is an important task [5]. However, PDEs are commonly hard to tackle, and their fractional order types are more complicated [2, 6–10]. Therefore, several methods such as the homotopy perturbation method [11], sub-ODE method [12, 13], generalized tanh method [14], residual power series method [14, 15], and so on [16–21] are developed to obtain solutions of some nonlinear fractional differential equations. Most of these methods sometimes require complex and huge calculation in order to obtain approximate solutions. To overcome such difficulties and drawbacks, an alternative method, the so called the fractional reduced differential transform method (FRDTM), has been developed by Keskin and Oturanc [22]. FRDTM plays a vital role among all the listed methods because it takes small size computation, easy to implement as compared to other techniques [23]. Using this method, it is possible to find both exact and approximate solutions in a rapidly convergent power series form. FRDTM is a very reliable, efficient, and effective powerful computational technique for solving physical problems [24–26].In the field of modern science and engineering, the fourth-order initial/boundary value problems are of great importance. For example, airplane wings, bridge slabs, floor systems, and window glasses are being modeled as plates subjected to different types of end supports which are successfully described in terms of fourth-order PDEs [27, 28]. Time-fractional NLPDEs come from classical NLPDEs by replacing its time derivative with a fractional derivative. Time-fractional partial differential equations have many applications in areas such as electrochemistry, diffusion processes, electromagnetic, material science, chaotic dynamics, and so forth [27, 29].In this work, we consider the class of three-dimensional fourth-order time-fractional parabolic PDEs of the form(1)∂2αu∂t2α+μx,y,z∂4u∂x4+λx,y,z∂4u∂y4+ηx,y,z∂4u∂z4=gx,y,z,a<x,y,z<b,t>0.Subjected to the initial conditions,(2)ux,y,z,0=f0x,y,z,∂∂tux,y,z,0=f1x,y,z.And boundary conditions(3)ua,y,z,t=g0y,z,t,ub,y,z,t=g1y,z,t,ux,a,z,t=k0x,z,t,ux,b,z,t=k1y,z,t,ux,y,a,t=k0x,y,t,ux,y,b,t=k1x,y,t,∂2ua,y,z,t∂x2=g¯0y,z,t,∂2ub,y,z,t∂x2=g¯1y,z,t,∂2ux,a,z,t∂y2=k¯0x,z,t,∂2ux,b,z,t∂y2=k¯1y,z,t,∂2ux,y,a,t∂z2=k¯0x,y,t,∂2ux,y,b,t∂z2=k¯1x,y,t,where 0<α≤1, μx,y,z, λx,y,z, and ηx,y,z are positive, and the functions fi,gi,ki,hi,f¯i,g¯i,k¯i,h¯i for i=0,1.The remaining sections of this study are organized as follows: in Section2, we give some fundamental definitions and lemmas associated with fractional calculus. In Section 3, some basic definitions and properties related to the three-dimensional fractional reduced differential transform method taking the fractional derivatives in the Caputo sense are presented, and some lemmas are proved. In Section 4, we present the formulation of the method. Section 5 is devoted to apply the method to solve three-dimensional time-fractional parabolic partial differential equation with variable coefficients. In Section 6, we present the numerical results and discussion. Tables and graphs are used to show the effectiveness, validity, and performance of the FRDTM for some values of α. Finally, the conclusion is presented in Section 7.


## 2. Fractional Calculus

In this section, some basic definitions and lemmas associated with fractional calculus are presented. Some of these definitions are due to Riemann–Liouville and Caputo sense; for details, see [2, 24, 30–33].Definition 1.
The gamma function. The gamma functionΓz is simply a generalization of the factorial real arguments. The Gamma function can be defined as(4)Γz=∫0∞e−ttz−1dt,z∈ℂ.Definition 2.
The fractional derivative offt in Caputo sense is defined as(5)Dtαft=Jn−αDnft=1Γn−α∫0tt−τn−α−1fnτdτ,n−1<α≤n,dndtnft,α=n.
The basic properties of the Caputo fractional derivative can be given by the following lemma.Lemma 1.
Ifm−1<α≤m,m∈ℕ and f∈Cμm,μ≥−1, then(6)DαJαfx=fx,x>0,DαJαfx=fx−∑k=0mfk0+xkk!,x>0.Definition 3.
Forp to be the smallest integer that exceeds α, and m to be the smallest integer that exceeds, β the Caputo time-fractional derivative of order α>0 is defined as(7)Dtαux,y,z,t=∂αux,y,z,t∂tα=1Γp−α∫0tt−ξp−α−1∂pux,y,z,ξ∂ξpdξ,p−1<α<p,∂αx,y,z,t∂tα,α=p∈N,and the space fractional derivative operator with respect to x, y, and z of order β>0 is, respectively, defined as(8)Dtβux,y,z,t=∂βux,y,z,t∂xβ=1Γn−β∫0tt−ξn−β−1∂nuβ,y,z,t∂ξndξ,n−1<β<n,∂βux,y,z,t∂xβ,β=n∈N,Dtβux,y,z,t=∂βux,y,z,t∂yβ=1Γm−β∫0tt−ξm−β−1∂mux,ξ,z,t∂ξmdξ,m−1<β<m,∂βux,y,z,t∂yβ,β=m∈N,Dtβux,y,z,t=∂βux,y,z,t∂zβ=1Γm−β∫0tt−ξm−β−1∂mux,y,ξ,t∂ξmdξ,m−1<β<m,∂βx,y,z,t∂zβ,β=m∈N.


## 3. Fractional Reduced Differential Transform Method (FRDTM)

In this section, the basic definition and properties related to the three-dimensional fractional reduced differential transform method taking the fractional derivatives in the Caputo sense are presented.Definition 4.
(see [33, 34]). If wx,y,z,t is analytic and continuously differentiable with respect to space variables x, y, and time variable t in the domain of interest, then the spectrum function is defined as(9)RDwx,y,z,t≈Wkx,y,z=1Γkα+1∂kα∂tkαux,y,z,tt=t0.Definition 5.
(see [33, 34]). The inverse FRDTM of Wkx,y,z is defined as(10)RD−1Wkx,y,z≈wx,y,z,t=∑k=0∞1Γkα+1Wkx,y,zt−tokα,wx,y,z,t=∑k=0∞1Γkα+1∂k∂tkwx,y,z,tt=t0.
The inverse transform of the set of values ofUkx,y,zk=0n gives the n-terms approximate solution as follows:(11)u¯nx,y,z,t=∑k=0nUkx,y,ztαk.
Therefore, the exact solution of the problem is given by(12)ux,y,z,t=limn⟶∞u¯nx,y,z,t=∑k=0∞Ukx,y,ztαk.Lemma 2.
Ifwx,y,z,t=fx,y,zux,y,z,t, then Wkx,y,z=fx,y,zUkx,y,z.Proof.
By Definition3 and properties of FRDTM given in Table 1, we have(13)Wkx,y,z=1Γkα+1∂kα∂tkαwx,y,z,tt=t0,Wkx,y,z=1Γkα+1∂kα∂tkαfx,y,zux,y,z,tt=t0,Wkx,y,z=fx,y,z1Γkα+1∂kα∂tkαux,y,z,tt=t0,Wkx,y,z=fx,y,zUkx,y,z.Table 1
The basic properties of FRDTM are described below [33, 34].
The original functionTransformed formwx,y,z,t=αux,y,z,t±βvx,y,z,tWkx,y,z=αUkx,y,z+βVkx,y,zwx,y,z,t=∂Nα/∂tNαux,y,z,tWkx,y,z=Γαk+N+1/Γkα+1Uk+Nx,y,zwx,y,z,t=fx,y,zux,y,z,tWkx,y,z=fx,y,zUkx,y,zwx,y,z,t=sinαx+βy+μz+ωtWkx,y,z=ωk/k!sinαx+βy+μz+πk/2, α,β,μ, and ω are the constantswx,y,z,t=cosαx+βy+μz+ωtWkx,y,z=ωk/k!cosαx+βy+μz+πk/2, α,β,μ, and ω are the constantsLemma 3.
Ifwx,y,z,t=fx,y,zsinαx+βy+μz+ωt, then Wkx,y,z=fx,y,zωk/k!sinαx+βy+μz+πk/2. α,β,μ, and ω are the constants.Proof.
By Definition3 and properties of FRDTM given in Table 1, we have(14)Wkx,y,z=1Γkα+1∂kα∂tkαwx,y,z,tt=t0,Wkx,y,z=1Γkα+1∂kα∂tkαfx,y,zsinαx+βy+μz+ωtt=t0,Wkx,y,z=fx,y,z1Γkα+1∂kα∂tkαsinαx+βy+μz+ωtt=t0,Wkx,y,z=fx,y,zωk/k!sinαx+βy+μz+πk/2.α,β,μ, and ω are the constants.Lemma 4.
Ifwx,y,z,t=fx,y,zcosαx+βy+μz+ωt, then Wkx,y,z=fx,y,zωk/k!cosαx+βy+μz+πk/2. α,β,μ, and ω are the constants.
The proof is similar to Lemma3.
Ifα=β=μ=0, then Lemma 4 reduces to Wkx,y,z=fx,y,zωk/k!cosπk/2, where k=0,1,2,….


## 4. Solution of the Problem by FRDTM

The steps to find the general solution of the governing equation (1) subjected to initial conditions (2) by the proposed method are as follows:Applying the properties of FRDTM given in Table1 to equations (1) and (2), we obtain(15)Γαk+2+1Γkα+1Uk+2x,y,z+μx,y,z∂4Ukx,y,z∂x4+λx,y,z∂4Ukx,y,z∂y4+ηx,y,z∂4Ukx,y,z∂z4=Gkx,y,z,(16)U0x,y,z=f0x,y,z,U1x,y,z=f1x,y,z.Solving forUk+2x,y,z from equation (15), we get(17)Uk+2x,y,z=−Γkα+1Γαk+2+1μx,y,z∂2Ukx,y,z∂x2+λx,y,z∂2Ukx,y,z∂y2+ηx,y,z∂4Ukx,y,z∂z4−Gkx,y,z.Whenk = 0,1,2,3,... by iterative calculations, we obtain the following successive terms:(18)U2x,y,z=−1Γ2α+1μx,y,z∂4f0x,y,z∂x4+λx,y,z∂4f0x,y,z∂y4+ηx,y,z∂4f0x,y,z∂z4−G0x,y,z,U3x,y,z=−Γα+1Γ3α+1μx,y,z∂2f1x,y,z∂x2+λx,y,z∂2f1x,y,z∂y2+ηx,y,z∂4f1x,y,z∂z4−G1x,y,z,and so on.Thus, the inverse transform of the set of values ofUkx,y,zk=0n giving the n-terms approximate solution unx,y,z,t=∑k=0nUkx,y,ztαk and the FRDTM solution of equation (1) subjected to the initial condition (2) is given by(19)ux,y,z,t=limn⟶∞unx,y,z,t=∑k=0∞Ukx,y,ztαk=fox,y,z+f1x,y,ztα+−1Γ2α+1μx,y,z∂4fox,y,z∂x4+λx,y,z∂4fox,y,z∂y4ηx,y,z∂4fox,y,z∂z2−Gox,y,zt2α+⋯.


## 5. Illustrative Examples

Example 1.
Consider the following partial differential equation in three space variables(20)∂2αu∂t2α+y+z2cosx−1∂4u∂x4+z+x2cosy−1∂4u∂y4+x+y2cosz−1∂4u∂z4=0,0<x,y,z<π3,0<α≤1,t>0,subjected to the initial conditions(21)ux,y,z,0=x+y+z−cosx+cosy+cosz,∂u∂tx,y,z,0=cosx+cosy+cosz−x+y+z.
Whenα=1, equation (20) reduces to the classical (or nonfractional) three-dimensional homogeneous parabolic partial differential equation [35–37].
Applying the properties of FRDTM given in Table1 and Lemma 2 to equations (20) and (21), we get(22)Uk+2=−Γkα+1Γαk+2+1y+z2cosx−1∂4Uk∂x4+z+x2cosy−1∂4Uk∂y4+x+y2cosz−1∂4Uk∂z4,U0x,y,z=x+y+z−cosx+cosy+cosz,U1x,y,z=cosx+cosy+cosz−x+y+z.
Whenk = 0,1,2,…, we have(23)U2x,y,z=1Γ2α+1x+y+z−cosx+cosy+cosz,U3x,y,z=−Γα+1Γ3α+1x+y+z−cosx+cosy+cosz,U4x,y,z=1Γ4α+1x+y+z−cosx+cosy+cosz,U5x,y,z=−Γα+1Γ5α+1x+y+z−cosx+cosy+cosz,and so on. Continuing in this way, the remaining steps of the FRDTM can be obtained. Then, by equation (19), we obtain the following approximate solution:(24)ux,y,z,t=x+y+z−cosx+cosy+cosz1−tα+1Γ2α+1t2α−Γα+1Γ3α+1t3α+1Γ4α+1t4α−⋯.
The exact solution of the classical form of Example1 is ux,y,z,t=x+y+z−cosx−cosy−cosze−t as in [35–37].Example 2.
Consider the following three-dimensional nonhomogeneous fourth-order parabolic partial differential equation:(25)∂2αu∂t2α+14!z∂4u∂x4+14!x∂4u∂y4+14!y∂4u∂z4=−xy−yz−zx+1x5+1y5+1z5cost,12<x,y,z<1,0<α≤1,t>0,subjected to initial conditions(26)ux,y,z,0=xy+yz+zx,∂∂tux.y,z,0=0.
Whenα=1, equation (25) reduces to the classical (or nonfractional) three-dimensional nonhomogeneous parabolic partial differential equation [37].
Applying properties of FRDTM given in Table1 and Lemma 4 to equations (25) and (26), we get(27)Γαk+2+1Γαk+1Uk+2x,y,z+14!z∂4Ukx,y,z∂x4+14!x∂4Ukx,y,z∂y4+14!y∂4Ukx,y,z∂z4=−xy−yz−zx+1x+1y+1z1k!cosπ2k,(28)U0x,y,z=xy+yz+zx,U1x.y,z=0.
Solving forUk+2x,y,z from equation (27), we have(29)Uk+2x,y,z=−Γαk+1Γαk+2+114!z∂4Ukx,y,z∂x4+14!x∂4Ukx,y,z∂y4+14!y∂4Ukx,y,z∂z4−−xy−yz−zx+1x5+1y5+1z51k!cosπ2k.
Now, whenk = 0, 1,3,…,(30)U2x,y,z=−1Γ2α+1xy+yz+zx,U3x,y,z=0,U4x,y,z=−Γ2α+1Γ4α+1−1Γ2α+11x5+1y5+1z5+−xy−yz−zx+1x5+1y5+1z512!,U5x,y,z=0,U6x,y,z=−Γ4α+1Γ6α+11Γ4α+170x9z+−Γ2α+12Γ4α+1−1x5+70x9z+1Γ4α+170y9x+−Γ2α+12Γ4α+1−1y5+70y9x+1Γ4α+170z9y+−Γ2α+12Γ4α+1−1z5+70z9y−−xy−yz−zx+1x5+1y5+1z514!,and so on. Continuing in this way, the remaining steps of the FRDTM can be obtained. Then, by equation (19), we obtain(31)ux,y,z,t=xy+yz+zx−xy+yz+zx1Γ2α+1t2α+−Γ2α+1Γ4α+1−1Γ2α+11x5+1y5+1z5+−xy−yz−zx+1x5+1y5+1z512!t4α+⋯.
The exact solution of the classical form of Example2 is ux,y,z,t=x/y+y/z+z/x1−1/2!t2+1/4!t4−⋯=x/y+y/z+z/xcost as in [37].


## 6. Numerical Results and Discussion

In this section, the numerical values of the functionu (x, y, z, t) of Examples 1 and 2 are computed for various values of the fractional order α, and the exact solutions for the variables x, y, z, and t are presented in tables and figures.Tables2 and 3 exhibit the behavior of FRDTM solutions for different values of fractional order α and the absolute errors (E) approximated in the 5th and 6th order at different values of the independent variables x, y, z, and time t of equations (20) and (25), respectively. As it can be seen from these tables, when the value of the fractional order α becomes close to 1, the obtained FRDTM solutions of Examples 1 and 2 are close to the exact solutions of its respective classical (nonfractional) equations obtained by different methods. In particular, when α = 1, the FRDTM solutions of the mentioned examples are in excellent agreement with the exact solutions of their corresponding classical (nonfractional) form, for details, see [35–37].Table 2
Fifth-order approximate numerical solution of equation (20) at different values of the fractional order α and the absolute error E = |uexact–uα = 1 | for u (x, y, z, t) by FRDTM.
xyztFRDTM for different values ofαuexactError (E)α = 0.5α = 0.7α = 0.9α = 10.10.30.20.3−1.57206−1.61233−1.68227−1.72641−1.7264100.5−1.47993−1.43721−1.41053−1.41342−1.413464E − 050.80.3−0.97616−1.00116−1.04459−1.072−1.07200.5−0.918948−0.892426−0.875856−0.877649−0.8776793E − 050.40.60.20.3−1.02973−1.05611−1.10192−1.13083−1.1308300.5−0.969381−0.941403−0.923924−0.925816−0.9258473.1E − 050.80.3−0.43383−0.444942−0.464243−0.476422−0.4764231E − 060.5−0.408403−0.396616−0.389252−0.390049−0.3900621.3E − 050.70.90.20.3−0.382167−0.391956−0.408958−0.419687−0.41968700.5−0.359768−0.349384−0.342897−0.3436−0.3436111.1E − 050.80.30.2137370.2192120.2287210.2347210.2347221E − 060.50.201210.1954030.1917750.1921670.1921747E − 06110.20.30.09398950.0963970.1005790.1032170.10321700.50.08848080.08592710.08433170.08450440.08450722.8E − 060.80.30.6898930.7075650.7382580.7576250.7576261E − 060.50.6494590.6307140.6190040.6202710.6202922.1E − 05Table 3
Sixth-order approximate numerical solution of equation (25) at different values of the fractional order α and the absolute error E = |uexact − uα = 1 | for u (x, y, z, t) by FRDTM.
xyztFRDTM for different values ofαuexactError (E)α=0.5α=0.7α=0.9α=10.60.650.60.5−287.047−15.13442.096352.638372.6383701−2317.74−321.166−16.96641.62431.624377E − 050.80.52.333122.505592.63872.693222.69322011.658061.658061.658061.658061.658148E − 050.70.750.60.52.31152.482372.614252.668272.66827011.64271.64271.64271.64271.642788E − 050.80.52.291142.460512.591222.644762.64476011.628231.628231.628231.628231.62837E − 050.80.850.60.52.362722.537382.672172.727392.72739011.67911.67911.67911.67911.679177E − 050.80.52.283522.452332.582612.635972.63597011.622821.622821.622821.622821.622897E − 050.90.90.60.52.407442.58542.722752.779012.77901011.710881.710881.710881.710881.710968E − 050.80.52.291292.460672.591392.644942.64494011.628341.628341.628341.628341.628417E − 05Figures1(a)–1(d) show the evaluation results of the approximate analytical solution of equation (20). These figures also show the behavior of the approximate solution obtained by the proposed method at fixed values x = 0.6 and y = 0.7 and for different values of the fractional order α = 0.5, 0.7, 1 and the exact solution.Figure 1
3D plots of FRDTM solutions of equation (20) for the different fractional order α and the exact solution.
(a)(b)(c)(d)Figures2(a) and 2(b) demonstrate the solutions u (x, y, z, t) verses t of equation (20) for a fixed value of x, y, and z. In Figure 2(a), the FRDTM solutions at α = 0.7, 0.8, 1 and the exact solution are compared for a fixed value x = y = z = 0.4. In Figure 2(b), the FRDTM solutions at α = 0.7, 0.8, 1 and the exact solution are compared for a fixed value x = y = z = 0.6.Figure 2
2D plots of FRDTM solutions of equation (20) for the different fractional order α and the exact solution.
(a)(b)Figures3(a)–3(d) show the evaluation results of the approximate analytical solution of equation (25). These figures also depict the behavior of the approximate solutions obtained by the proposed method at fixed values x=0.6 and y=0.7 and for a different time-fractional order α=0.5,0.7,1 and the exact solution.Figure 3
3D plots of FRDTM solutions of equation (25) for the different fractional order α and the exact solution.
(a)(b)(c)(d)Figures4(a) and 4(b) demonstrate the solutions ux,y,z,t verses t of equation (25) for a fixed value of x,y,z. In Figure 4(a), the FRDTM solutions at α=0.8,0.9,1 and exact solution are compared for a fixed value x=y=z=0.8. In Figure 4(b), the FRDTM solutions at α=0.8,0.9,1 and exact solution are compared for a fixed value x=y=z=0.9.Figure 4
2D plots of FRDTM solutions of (25) for the different fractional order α and the exact solution.
(a)(b)


## 7. Conclusion

In this study, we successfully employed the FRDTM to find the approximate and exact solutions of three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients taking the fractional derivative in Caputo sense. It is found that the results we obtained in Examples1 and 2 when α=1 are in excellent agreement with the exact solutions obtained by VIM [35], ADM [36], and RDTM [37]. Also, the illustrated examples demonstrate that the suggested technique can be observed as a fast, simple, and efficient tool for computing three-dimensional fourth-order time-fractional parabolic partial differential equations (PDEs). The main advantage of the method is that it can be applied to fractional PDE’s and obtain approximate and exact solutions of fractional PDEs without requiring any discretization, perturbation, or restrictive conditions. Moreover, the tables and graphs constructed in this work demonstrate the physical behavior of the FRDTM solutions of the problems for different values of α. As a result, we conclude that the FRDTM is very powerful, straightforward, and effective to obtain approximate and exact solutions of a wide variety problems related to fractional PDEs applied in sciences such as mathematics, physics, and engineering.



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