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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Semi- analytic numerical method for solution of time-space fractional heat and wave type equations with variable coefficients

Rishi Kumar Pandey / Hradyesh Kumar Mishra
Published Online: 2017-03-17 | DOI: https://doi.org/10.1515/phys-2017-0009

Abstract

The time and space fractional wave and heat type equations with variable coefficients are considered, and the variable order derivative in He‘s fractional derivative sense are taken. The utility of the homotopy analysis fractional sumudu transform method is shown in the form of a series solution for these generalized fractional order equations. Some discussion with examples are presented to explain the accuracy and ease of the method.

Keywords: He‘s fractional derivative; Fractional partial differential equation; Heat equations; Homotopy Analysis Fractional Sumudu Transform Method; Wave equation

PACS: 02.20.Sv; 47.15.–x; 44.40.+a; 44.10.+i

1 Introduction

The partial differential equations involving the variable order derivatives provide the more accurate results, instead of integer order conventional derivatives in many physical applied mathematical models. In the last two decades many mathematicians and scientist have shown great interest in the use of fractional derivatives in many senses like caputo [1, 2], Riemann–Liouville [1], Riesz derivative [3], Weyl derivative[4] etc. Some basics of fractional differential equations have been discussed in engineering and physics [1],[57].

Many fields are conventionally using the fractional partial differentiation and fractional integration, and few are in initial stage. Some applications can be found in image processing, plasma physics, nonlinear control theory, biological modelling, and astrophysics etc. [816]. Hu et al. [17] provide the fractal space-time scales observation based on a smaller threshold model in micro and nano scale with smoothness. The practical aspects of the model are discussed by Ji-Huan He in his fractional derivative. A space and time fractional wave and heat equations are the linear integro partial equations arising mainly in conventional diffusion or wave equations using the fractional derivative of arbitrary order instead of the conventional derivative [18]. The result of the space–time fractional wave equations have many applications. For example, to investigate the Brownian diffusion, unification of diffusion and propagation phenomena of a wave, sub –diffusion systems, and random walk [1922].

Some attempts are investigated to solve the multi–term space–time fractional derivatives by the Variational iteration method [23], Adomian decomposition method [24], Homotopy analysis method [25], and Homotopy perturbation method [26, 27].

The main purpose of the present manuscript is to apply the homotopy analysis fractional sumudu transform method (HAFSTM) [2830] to evaluate the discussed problem and obtain the approximate converging series solution. In this discussion, we also evaluate the solution for both domains of space and time with lucid manner implementation, which was not previously discussed in the available literature, and also incorporating the balance of convergence of the fractional term presented by graphs in numerical experiments.

2 Preliminaries and Notations

This section contains the brief portrayal of the active possession of the idea of obtaining the solutions with adequate theory of fractional calculus, which facilitate us to obtain the solution of the problem specified in this manuscript. Basic definition of, Riemann –Liouville, Caputo, derivatives and sumudu integral transform and expansion of fractional derivative using the transform, is also discussed.

Definition 1

The left sided Liouville Fractional integral operator of order α 0, of a function f (t) ∈Cµ, and µ –1is defined as [31,32] Jαft=1Γα0ttτα1fτdτ,α>0,x>0

and J0ft=ft.

Definition 2

Let the function f (t) , t > 0, be in the space Cµ, µ ∈ ℝ if there exists a real number p (> µ), such that f (t) = tpf1(t), where f1(t) ∈C [0, ∞), and it is said to be in the space Cμm if ƒ(m)Cµ, mN. [33]

Definition 3

The Riemann-Liouville fractional differential operator of order α ≥0, [1,30] Dαft=dmdtmImαft,m1<αm,mN.

Definition 4

The left sided caputo of f (t) derivative is defined as [1] Dtαft=JmαDnft,1Γnα0ttTmα1fmτdτ,

where m - 1 < αm, mN, t > 0.

Definition 5

Ji–Huan He‘s fractional derivative is defined as [34] αfttα=1Γnαntn0tτtnα1f0τfτdτ,n1<α<n.

Definition 6

In early 90‘s, Watugala introduced an incipient integral transform. The sumudu transform is defined over the set of functions [28, 35] A=ftM,τ1,τ2>0,ft<Metτj,ift1j×0,,

by the following formula f¯u=Sft=0futetdt,uτ1,τ2.

Definition 7

The sumudu transform of f (t) = tα is defined as [35] Stα=0ettαdt=Γα+1uα,Rα>0.

Definition 8

The Sumudu transform 𝕊[f(t)] of the Riemann-Liouville fractional integral is defined as [30,35] SIαft=uαFu.

Definition 9

The Sumudu transform 𝕊[f (t)] of the Caputo fractional derivative is defined as [30,36] SDtαft=uαSftk=0m1uα+kfk0+,

where m1<αm.

3 Analysis of the method

This section is devoted to deriving the algorithm for the space-time FPDE of heat and wave type at 0 < α ≤1, 1 < α ≤ 2 respectively.

We consider the following equation for the heat and wave form DtαUx,y,z,t=fx,y,zx2βUx,y,z,t+gx,y,zy2βx,y,z,t+hx,y,zz2γx,y,z,t,0<x<a,0<y<0,0<z<c,0<β1,0<γ1.(1)

Eq. (1) represents the heat equation when 0 < α ≤1, and wave equation for 0 < α ≤ 2.

Using the sumudu transform of Eq. (1) on both sides, we get SUx,y,z,tuαk=0n1Uk0uαk=Sfx,y,zx2βUx,y,z,t+gx,y,zy2βUx,y,z,t+hx,y,zz2γUx,y,z,t(2)

Now, we define nonlinear operator as Nφx,y,z,t;q=Sφx,y,z,t;qk=0n1Uk0ukuαSfx,y,zx2βφx,y,z,t;q+gx,y,zy2βφx,y,z,t;q+hx,y,zz2γφx,y,z,t;q,(3)

where q ∈ [0,1] is an embedding parameter and φ (x, y, z, t;q) is a real function of x, y, z, t, and q. we construct a homotopy as follow: 1qSφx,y,z,t;qU0x,y,z,t=qHx,y,z,tNφx,y,z,t;q(4)

where ℏ is a nonzero auxiliary parameter and H(x, y, z, t) ≠ 0. An auxiliary function U0(x,y, z,t) is an initial guess of U (x, y, z, t), and φ (x, y, z, t; q) is an unknown function. It is important that one has great freedom to choose the auxiliary parameter in HAFSTM. Obviously, when q = 0 and q = 1 it holds [36] φx,y,z,t;0=U0x,y,z,t,φx,y,z,t;1=Ux,y,z,t(5)

Consequently, while q increases from 0 to 1, the solution converges from initial deduction U0 (x, y, z, t) to the solution U (x, y, z, t) . Now, expanding φ (x, y, z, t;q) on Taylor’s series with respect to q, we get [36] φx,y,z,t;q=U0x,y,z,t+m=1qmUmx,y,z,t(6)

where Umx,y,z,t=1m_mφx,y,z,t;qqmq=0(7)

The convergence of the series solution (6) is steering through ℏand stipulation of initial guess, the auxiliary linear operator, and the auxiliary function. The series (6) converges at q = 1. Hence we obtain [36] Ux,y,z,t=U0x,y,z,t+m=1Umx,y,z,t(8)

which must be one of the solutions of the original nonlinear equations. The above expression provides us with an association between the initial guess U0 (x, y, z,t) and the exact solution U (x, y, z,t) by means of the terms Um (x, y, z,t) (m = 1, 2, 3, ...) , which are still to be determined.

Define the vectors U=U0x,y,z,t,U1x,y,z,t,...,Umx,y,z,t.(9)

Differentiating the zero order deformation Eq. (4) m times with respect to embedding parameter q and then setting q = 0, and finally dividing them by m!, we obtain the mth order deformation equation as follows: SUmx,y,z,tχmUm1x,y,z,t=Hx,y,z,tRmUm1,x,y,z,t.(10)

Operating the inverse Sumudu transform on both sides, we get Umx,t=χmUm1x,y,z,t+S1Hx,y,z,tRmUm1,x,y,z,t,(11)

where RmUm1,x,y,z,t=1m1!m1φx,y,z,t;qqm1q=0(12)

and χm=0,m1,1m>1.

In this way, it is straightforward to acquire Um (x, y, z,t) for m ≥ 1. At Nth order, we have Ux,y,z,t=m=0NUmx,y,z,t,(13)

where N → ∞, we obtain an accurate approximation of the original equation (1).

4 Illustrative Examples

To demonstrate the effectiveness and the precision of the above discussed method. Here, we apply the HAFSTM to solve some space –time fractional wave and heat type equations.

Example 4.1

Consider the following one-dimensional space –time fractional heat-like problem DtαUx,t=12x2x2βUx,t,0<x<1,0<α1,0<β1,t>0,(14)

subject to the boundary conditions U (0, t) = 0, U (1, t) = et , and the initial condition Ux,0=x2.

Applying the sumudu transform on both sides in Eq. (14), SDtαUx,tS12x2x2βUx,t=0, SDtαUx,tk=0n1Uk0ukuαS12x2x2βUx,t=0,

The nonlinear operator is defined by Nφx,t;q=Sφx,t;qk=0n1φk0ukuαS12x2x2βφx,t;q,(15)

and thus RmUm1,x,t=SUm1x,t+uαS12x2x2βUm1x,t.(16)

The mth- order deformation equation is given by SUmx,tχmUm1x,t=Hx,tRmUm1,x,t.

Applying the inverse Sumudu transform, we have Umx,t=χmUm1x,t+S1Hx,tRmUm1,x,t.(17)

Solving Eq. (17) by using Eq. (16) for m = 1, 2, ..., we obtain U1x,t=x42βΓ42βtα+1Γα+2 U2x,t=x42βΓ42βtα+1Γα+22x42βΓ42βtα+1Γα+2+22Γ52βx42βΓ42βΓ54βt2α+1Γ2α+2 U3x,t=x42βΓ42βtα+1Γα+22x42βΓ42βtα+1Γα+2+22Γ52βx42βΓ42βΓ54βt2α+1Γ2α+2+x42βΓ42βtα+1Γα+22x42βΓ42βtα+1Γα+2+22Γ52βx42βΓ42βΓ54βt2α+1Γ2α+222Γ52βx64βΓ42βΓ54βt2α+1Γ2α+2+32Γ52βx64βΓ42βΓ54βt2α+1Γ2α+232Γ52βΓ74βx64βΓ42βΓ54βΓ76βt3α+1Γ3α+2

etc.. In the same manner, the rest of the components of (14) as a series m ≥ 4 can be obtained.

The solution of (14) is given by Ux,t=U0x,t+m=1Umx,t.(18)

The accuracy and convergence of the HAFSTM series solution depends on the careful selection of the auxiliary parameter ℏ. Here, we choose ℏ= -1, then Ux,t=x2+3tαx42βΓα+1Γ32β+t2αx64βΓ52β2Γ2α+1Γ54βΓ32β+t3αx86βΓ72βΓ52β4Γ3α+1Γ76βΓ54βΓ32β+.....(19)

For β = 1, Eq. (19) reduces to exactly the same as given in the Decomposition method by S. Momani [37], and also converges to the exact solution when α = β = 1, Ux,t=x2et(20)

obtained by Ozis [38] and Wazwaz [39] for standard heat type equation

Example 4.2

Consider the following two-dimensional space –time fractional heat-like problem DtαUx,y,t=x2βUx,y,t+y2γUx,y,t,0<x,y<2π,0<α1,0<β1,0<Γ1,t>0,(21)

subject to the boundary conditions U0,y,t=0,U2π,y,t=0,Ux,0,t=0,Ux,2π,t=0,

Plot of the U (x, t) when α = 0.976, β = 0.849 green curve: approximate solution; red curve: exact solution at x = 0.572 and ℏ= -0.87.
Figure 1

Plot of the U (x, t) when α = 0.976, β = 0.849 green curve: approximate solution; red curve: exact solution at x = 0.572 and ℏ= -0.87.

and the initial condition Ux,y,0=sinxsiny,

Applying the sumudu transform on both sides in Eq. (21), SDtαUx,y,tSx2βUx,y,t+y2γUx,y,t=0, SDtαUx,y,tk=0n1Uk0ukuαSx2βUx,y,t+y2γUx,y,t=0.

The nonlinear operator is defined by Nφx,y,t;q=Sφx,y,t;qk=0n1φk0ukuαSx2βφx,y,t;q+y2γφx,y,t;q,(22)

and thus RmUm1,x,y,t=SUm1x,y,t+uαSx2βUm1x,y,t+y2γUm1x,y,t.(23)

The mth- order deformation equation is given by SUmx,y,tχmUm1x,y,t=Hx,y,tRmUm1,x,y,t.

Applying the inverse Sumudu transform, we have Umx,y,t=χmUm1x,y,t+S1Hx,y,tRmUm1,x,y,t.(24)

Solving Eq. (24) using Eq. (23) for m = 1, 2, ..., we obtain U1x,y,t=i=11ix2i+12βΓ2i+22βj=01jy2i+1Γ2j+2+i=11ix2i+1Γ2i+2j=01jy2i+12γΓ2j+22γtαΓα+1, U2x,y,t=i=11ix2i+12βΓ2i+22βj=01jy2j+1Γ2j+2+i=01ix2i+1Γ2i+2j=11jy2j+12γΓ2j+22γtαΓα+12i=11ix2i+12βΓ2i+22βj=01jy2j+1Γ2j+2+i=01ix2i+1Γ2i+2j=11jy2j+12γΓ2j+22γtαΓα+1,+2i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+1++2i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+1

U3x,y,t=i=11ix2i+12βΓ2i+22βj=01jy2j+1Γ2j+2+i=01ix2i+1Γ2i+2j=11jy2j+12γΓ2j+22γtαΓα+12i=11ix2i+12βΓ2i+22βj=01jy2j+1Γ2j+2+i=01ix2i+1Γ2i+2j=11jy2j+12γΓ2j+22γtαΓα+1+2i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+1+2i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+1+i=11ix2i+12βΓ2i+22βj=01jy2j+1Γ2j+2+i=01ix2i+1Γ2i+2j=11jy2j+12γΓ2j+22γtαΓα+1+2i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+1+2i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+1+2i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+1+2i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+1+3i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+13i=31ix2i+16βΓ2i+26βj=01jy2j+1Γ2j+2 +i=21ix2i+14βΓ2i+24βj=11jy2j+12γΓ2j+22γ+i=11ix2i+12βΓ2i+22βj=21jy2i+14γΓ2j+24γt3αΓ3α+1+2i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+1+3i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+13i=21ix2i+14βΓ2i+24βj=11jy2j+12γΓ2j+22γ+i=11ix2i+12βΓ2i+22βj=21jy2i+14γΓ2j+24γ+i=11ix2i+12βΓ2i+22βj=21jy2j+14γΓ2j+24γ+i=01ix2i+1Γ2i+2j=31jy2i+16γΓ2j+26γt3αΓ3α+1+2i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+1+2i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+1+2i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+1+3i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+13i=31ix2i+16βΓ2i+26βj=01jy2j+1Γ2j+2+i=21ix2i+14βΓ2i+24βj=11jy2i+12γΓ2j+22γ+i=21ix2i+14βΓ2i+24βj=11jy2j+12γΓ2j+22γ+i=11ix2i+12βΓ2i+22βj=21jy2i+14γΓ2j+24γt3αΓ3α+1

+2i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+1+3i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+13i=21ix2i+14βΓ2i+24βj=11jy2j+12γΓ2j+22γ+i=11ix2i+12βΓ2i+22βj=21jy2i+14γΓ2j+24γ+i=11ix2i+12βΓ2i+22βj=21jy2j+14γΓ2j+24γ+i=01ix2i+1Γ2i+2j=31jy2i+16γΓ2j+26γt3αΓ3α+1,

etc.. In the same way, the other components of (21) as a series m ≥ 4 can be obtained The solution of (21) is given by Ux,y,t=U0x,y,t+m=1Umx,y,t.(25)

The precision and convergence of the HAFSTM series solution depends on the useful choice of the auxiliary parameter ℏ. For convenience, we take ℏ= -1, then Ux,y,t=i=11ix2i+12βΓ2i+22βj=01jy2i+1Γ2j+2+i=11ix2i+1Γ2i+2j=01jy2i+12γΓ2j+22γtαΓα+1+i=21ix2i+14βΓ2i+24βj=01jy2j+1Γ2j+2+i=11ix2i+12βΓ2i+22βj=11jy2i+12γΓ2j+22γt2αΓ2α+1+i=11ix2i+12βΓ2i+22βj=11jy2j+12γΓ2j+22γ+i=01ix2i+1Γ2i+2j=21jy2i+14γΓ2j+24γt2αΓ2α+1+....(26)

For γ = β = 1, Eq. (26) reduces to exactly the same as given in the Decomposition method by S. Momani [37], and also converges to the exact solution when α = β = γ = 1, Ux,y,t=e2tsinxsiny(27)

obtained by Ozis [38] and Wazwaz [39] for standard heat type equation.

Plot of the U(x,y, t)when α = 0.84, β = 0.868, γ = 0.986 green curve: approximate solution; red curve: exact solution at x = 4.57416, y = 5.21504 and ℏ= -0.38.
Figure 2

Plot of the U(x,y, t)when α = 0.84, β = 0.868, γ = 0.986 green curve: approximate solution; red curve: exact solution at x = 4.57416, y = 5.21504 and ℏ= -0.38.

Example 4.3

Consider the following three-dimensional space -time fractional heat-like problem DtαUx,y,z,t=x4y4z4+136x2x2βUx,y,z,t+y2y2γUx,y,z,t+z2z2δUx,y,z,t,0<x,y,z<1,0<α1,0<β1,0<γ1,0<δ1,t>0,(28)

subject to the boundary conditions U0,y,z,t=0,U1,y,z,t=y4z4et1,Ux,0,z,t=0,Ux,1,z,t=x4z4et1,Ux,y,0,t=0,Ux,y,1,t=x4y4et1

and the initial condition Ux,y,z,0=0.

Applying the sumudu transform on both sides in Eq. (28), SDtαUx,y,tSx4y4z4+136x2x2βUx,y,z,t+y2y2γUx,y,z,t+z2z2δUx,y,z,t=0,

SDtαUx,y,tk=0n1Uk0ukuαSx4y4z4+136x2x2βUx,y,z,t+y2y2γUx,y,z,t+z2z2δUx,y,z,t=0.

The nonlinear operator is defined by Nφx,y,z,t;q=φx,y,z,t;qk=0n1φk0ukuαx4y4z4+136x2x2βφx,y,z,t;q+y2y2γφx,y,z,t;q+z2z2δUφx,y,z,t;q,(29)

and thus RmUm1,x,y,z,t=SUm1x,y,z,t+uαS(1χm)x4y4z4+136x2x2βUx,y,z,t+y2y2γUx,y,z,t+z2z2δUx,y,z,t.(30)

The mth- order deformation equation is given by SUmx,y,z,tχmUm1x,y,z,t=Hx,y,z,tRmUm1,x,y,z,t.

Applying the inverse Sumudu transform, we have Umx,y,z,t=χmUm1x,y,z,t+S1Hx,y,z,tRmUm1,x,y,z,t.(31)

Solving Eq. (31) using Eq.(30) for m = 1, 2, ..., we get U1x,y,z,t=x4y4z4tαΓα+1, U2x,y,z,t=x4y4z4tαΓα+12x4y4z4tαΓα+1+236Γ5x62βy4z4Γ52β+Γ5x4y62Γz4Γ52γ+Γ5x4y4z62δΓ52δt2αΓ2α+1 U3x,y,z,t=x4y4z4tαΓα+12x4y4z4tαΓα+1+236Γ5x62βy4z4Γ52β+Γ5x4y62γz4Γ52γ+Γ5x4y4z62δΓ52δt2αΓ2α+1+x4y4z4tαΓα+12x4y4z4tαΓα+1+236Γ5x62βy4z4Γ52β+Γ5x4y62γz4Γ52γ+Γ5x4y4z62δΓ52δt2αΓ2α+1+236Γ5x62βy4z4Γ52βt2αΓ2α+1+336Γ5x62βy4z4Γ52βt2αΓ2α+1236Γ5Γ72βx84βy4z4Γ74βΓ52β+Γ52x62βy62γz4Γ52βΓ52γ+Γ52x62βy4z62δΓ52βΓ52δt3αΓ3α+1+236Γ5x4y62γz4Γ52γt2αΓ2α+1+336Γ5x4y62γz4Γ52γt2αΓ2α+1236Γ52x62βy62γz4Γ52βΓ52γ+Γ5Γ72γx4y84γz4Γ52γΓ74γ+Γ52x4y62γz62δΓ52γΓ52δt3αΓ3α+1+236Γ5x4y4z62δΓ52δt2αΓ2α+1+336Γ5x4y4z62δΓ52δt2αΓ2α+1236Γ52x62βy4z62δΓ52βΓ52δ+Γ52x4y62γz62δΓ52γΓ52δ+Γ5Γ72δx4y84γz4Γ52δΓ74δt3αΓ3α+1

and so on. The other components of the series can easily be obtain by the iteration process. The solution of (28) is given by Ux,y,z,t=U0x,y,z,t+m=1Umx,y,z,t.(32)

The accuracy and convergence of the HAFSTM series solution depends on the careful selection of the auxiliary parameter ℏ. Here, we choose ℏ= -1, then Ux,y,z,t=tαx4y4z4Γ1+α+2t2αx62βy4z43Γ1+2αΓ52β+t3αx84βy4z4Γ72β54Γ1+3αΓ74βΓ52β+2t2αx4y62γz43Γ1+2αΓ52γ+8t3αx62βy62γz49Γ1+3αΓ52γΓ52β+t3αx4y84γz4Γ72γ54Γ1+3αΓ74γΓ52γ+2t3αx4y4z62δ3Γ1+3αΓ52δ+4t3αx62βy4z62δ9Γ1+3αΓ52βΓ52δ+4t3αx4y62γz62δ9Γ1+3αΓ52βΓ52δ+4t3αx4y62γz62δ9Γ1+3αΓ52γΓ52δ+4t3αx4y62γz62δ9Γ1+3αΓ52γΓ52δ+t3αx4y4z84δΓ72δ54Γ1+3αΓ74δΓ52δ(33)

For γ = β = δ = 1, Eq. (33) reduces to exactly the same as given in the Decomposition method by S. Momani [37], and also converges to the exact solution when α = β = γ = δ = 1, Ux,y,z,t=x4y4z4et1(34)

obtained by Ozis [38] and Wazwaz [39] for standard heat type equation.

Example 4.4

Consider the following one-dimensional space –time fractional wave-like problem DtαUx,t=12x2x2βUx,t,0<x<1,1<α2,0<β1,t>0,(35)

subject to the boundary conditions U (0, t) = 0, U (1, t) = 1 + sinh t, and the initial condition Ux,0=x,Utx,0=x2.

Applying the sumudu transform on both sides in Eq. (35), SDtαUx,tS12x2x2βUx,t=0, SDtαUx,tk=0n1Uk0ukuαS12x2x2βUx,t=0.

Plot of the U (x,y,z, t) when α = 0.806, β = 0.956, γ= 0.812, δ = 0.934 green curve: approximate solution; red curve: exact solution at x = 0.860, y = 1, z = 0.774 and ℏ= -1.925.
Figure 3

Plot of the U (x,y,z, t) when α = 0.806, β = 0.956, γ= 0.812, δ = 0.934 green curve: approximate solution; red curve: exact solution at x = 0.860, y = 1, z = 0.774 and ℏ= -1.925.

The nonlinear operator is defined by Nφx,t;q=Sφx,t;qk=0n1φk0ukuαS12x2x2βφx,t;q,(36)

and thus RmUm1,x,t=SUm1x,t+uαS12x2x2βUm1x,t.(37)

The mth- order deformation equation is given by SUmx,tχmUm1x,t=Hx,tRmUm1,x,t.

Applying the inverse Sumudu transform, we have Umx,t=χmUm1x,t+S1Hx,tRmUm1,x,t.(38)

Solving Eq. (38) using Eq.(37) for m = 1, 2, ..., we get U1x,t=x42βΓ42βtα+1Γα+2U2x,t=x42βΓ42βtα+1Γα+22x42βΓ42βtα+1Γα+2+22Γ52βx64βΓ42βΓ54βt2α+1Γ2α+2

U3x,t=x42βΓ42βtα+1Γα+22x42βΓ42βtα+1Γα+2+22Γ52βx42βΓ42βΓ54βt2α+1Γ2α+2+x42βΓ42βtα+1Γα+22x42βΓ42βtα+1Γα+2+22Γ52βx42βΓ42βΓ54βt2α+1Γ2α+222Γ52βx64βΓ42βΓ54βt2α+1Γ2α+2+32Γ52βx64βΓ42βΓ54βt2α+1Γ2α+232Γ52βΓ74βx86βΓ42βΓ54βΓ76βt3α+1Γ3α+2

etc.. In the same manner the rest of the components of (35) as a series m ≥ 4 can be obtained. The solution of (35) is given by Ux,t=U0x,t+m=1Umx,t.(39)

The accuracy and convergence of the HAFSTM series solution depends on the careful selection of the auxiliary parameter ℏ. Here, we choose ℏ= -1, then Ux,t=x2+tx2+tα+1x42βΓα+2Γ42β+t2α+1x64βΓ52β2Γ2α+2Γ54βΓ42β+t3α+1x86βΓ72βΓ52β4Γ3α+2Γ76βΓ54βΓ42β+.....(40)

For β = 1, Eq. (40) educes to exactly the same as given in the Decomposition method by S. Momani [37], and also converges to the exact solution when α = β = 1, Ux,t=x+x2sinht(41)

obtained by Ozis [38] and Wazwaz [39] for standard heat type equation.

Plot of the U(x, t)when α = 0.91, β = 0.582 green curve: approximate solution; red curve: exact solution at x = 0.445 and ℏ= -1.54.
Figure 4

Plot of the U(x, t)when α = 0.91, β = 0.582 green curve: approximate solution; red curve: exact solution at x = 0.445 and ℏ= -1.54.

Example 4.5

Consider the following three-dimensional space –time fractional wave-like problem DtαUx,y,z,t=x2+y2+z2+12x2x2βUx,y,z,t+y2y2γUx,y,z,t+z2z2δUx,y,z,t,0<x,y,z<1,1<α2,0<β1,0<Γ1,0<δ1,t>0,(42)

subject to the boundary conditions 0,y,z,t=y2et1+z2et1,U1,y,z,t=1+y2et1+z2et1,Ux,0,z,t=x2et1+z2et1,Ux,1,z,t=1+x2et1+z2et1,Ux,y,0,t=x2+y2et1,Ux,y,1,t=x2+y2et1+et1

and the initial condition Ux,y,z,0=0,Utx,y,z,0=x2+y2z2.

Applying the sumudu transform on both sides in Eq. (42), SDtαUx,y,tSx2+y2+z2+12x2x2βUx,y,z,t+y2y2γUx,y,z,t+z2z2δUx,y,z,t=0,

SDtαUx,y,tk=0n1Uk0ukuαSx2+y2+z2+12x2x2βUx,y,z,t+y2y2γUx,y,z,t+z2z2δUx,y,z,t=0.

The nonlinear operator is defined by Nφx,y,z,t;q=Sφx,y,z,t;qk=0n1φk0ukuαSx2+y2+z2+12x2x2βφx,y,z,t;q+y2y2γφx,y,z,t;q+z2z2δφx,y,z,t;q,(43)

and thus RmUm1,x,y,z,t=SUm1x,y,z,t+uαS1χmx2+y2+z2+12x2x2βUx,y,z,t+y2y2γUx,y,z,t+z2z2δUx,y,z,t(44)

The mth- order deformation equation is given by SUmx,y,z,tχmUm1x,y,z,t=Hx,y,z,tRmUm1,x,y,z,t.

Applying the inverse Sumudu transform, we have Umx,y,z,t=χmUm1x,y,z,t+S1Hx,y,z,tRmUm1,x,y,z,t.(45)

We start with U0x,y,z,t=x2+y2z2t+Jαx2+y2z2t,=x2+y2t+tαΓα+1+z2t+tαΓα+1

Solving Eq. (45) using Eq.(44) for m = 1, 2, ..., we get U1x,y,z,t=x2+y2t+tαΓα+1+z2t+tαΓα+1+x2+y2t+tαΓα+1+z2t+tαΓα+1+x42βΓ32β+y42γΓ32γtα+1Γα+2+t2αΓ2α+1z42δΓ32δtα+1Γα+2+t2αΓ2α+1,U2x,y,z,t=x2+y2t+tαΓα+1+z2t+tαΓα+1+x2+y2t+tαΓα+1+z2t+tαΓα+1x42βΓ32β+y42γΓ32γtα+1Γα+2+t2αΓ2α+1z42δΓ32δtα+1Γα+2+t2αΓ2α+1+x2+y2t+tαΓα+1+z2t+tαΓα+1+x2+y2t+tαΓα+1+z2t+tαΓα+1x42βΓ32β+y42γΓ32γtα+1Γα+2+t2αΓ2α+1z42δΓ32δtα+1Γα+2+t2αΓ2α+1x42βΓ32βtα+1Γα+2+t2αΓ2α+12x42βΓ32βtα+1Γα+2+t2αΓ2α+1+22Γ52βx64βΓ32βΓ54βt2α+1Γ2α+2+t3αΓ3α+1y42γΓ32γtα+1Γα+2+t2αΓ2α+12y42γΓ32γtα+1Γα+2+t2αΓ2α+1+22Γ52γy64γΓ32γΓ54γt2α+1Γ2α+2+t3αΓ3α+1z42δΓ32δtα+1Γα+2+t2αΓ2α+12z42δΓ32δtα+1Γα+2+t2αΓ2α+1+22Γ52δz64δΓ32δΓ54δt2α+1Γ2α+2+t3αΓ3α+1,

and so on. The other components of the series can easily be obtained by the iteration process.

The solution of (42) is given by Ux,y,z,t=U0x,y,z,t+m=1Umx,y,z,t.(46)

The accuracy and convergence of the HAFSTM series solution depends on the careful selection of the auxiliary parameter ℏ. Here, we choose ℏ= -1, then Ux,y,z,t=tαx4y4z4Γ1+α+2t2αx62βy4z43Γ1+2αΓ52β+t3αx84βy4z4Γ72β54Γ1+3αΓ74βΓ52β+2t2αx4y62γz43Γ1+2αΓ52γ+8t3αx62βy62γz49Γ1+3αΓ52γΓ52β+t3αx4y84γz4Γ72γ54Γ1+3αΓ74γΓ52γ+2t3αx4y4z62δ3Γ1+3αΓ52δ+4t3αx62βy4z62δ9Γ1+3αΓ52βΓ52δ+4t3αx4y62γz62δ9Γ1+3αΓ52βΓ52δ+4t3αx4y62γz62δ9Γ1+3αΓ52γΓ52δ+4t3αx4y62γz62δ9Γ1+3αΓ52γΓ52δ+t3αx4y4z84δΓ72δ54Γ1+3αΓ74δΓ52δ(47)

For γ = β = δ = 1, Eq. (47) reduces to exactly the same as given in the Decomposition method by S. Momani [37], and also converges to the exact solution when α = β = γ = δ = 1, Ux,y,z,t=x4y4z4et1(48)

obtained by Ozis [38] and Wazwaz [39] for standard heat type equation.

5 Concluding remark

In this article HAFSTM has been successfully applied for the solution of heat and wave type equations. The instant arbitrary graphs show how the arbitrary order of fractional derivatives also control the convergence rate corresponding to the exact solution. Apart from other methods, the HAFSTM provide highly convergent series solutions, and is also compatible with the application of the multi –term space –time fractional partial differential equations. In conclusion, the HAFSTM may be well thought-out as an elegant refinement of existing numerical methods, which may culminate in discoveries across the broad utility in science and engineering.

Plot of the U(x,y, z, t)when α = 0.923, β = 0.783, γ= 0.916, δ = 0.944 green curve: approximate solution; red curve: exact solution at x = 0.135, y = 0.182, z = 0.1andℏ = -2.13.
Figure 5

Plot of the U(x,y, z, t)when α = 0.923, β = 0.783, γ= 0.916, δ = 0.944 green curve: approximate solution; red curve: exact solution at x = 0.135, y = 0.182, z = 0.1andℏ = -2.13.

Acknowledgement

The authors are grateful to the referees for their valuable suggestions and comments for the improvement of the paper. The First author acknowledges the financial support provided by the JUET, Guna, India as a teaching assistantship.

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About the article

Received: 2016-09-14

Accepted: 2016-12-15

Published Online: 2017-03-17


Citation Information: Open Physics, Volume 15, Issue 1, Pages 74–86, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0009.

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© 2017 Rishi Kumar Pandey and Hradyesh Kumar Mishra. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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