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

  • Rishi Kumar Pandey and Hradyesh Kumar Mishra EMAIL logo
Published/Copyright: March 17, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

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],[5–7].

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. [8–16]. 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 [19–22].

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) [28–30] 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=1Γα∫0tt−τα−1fτdτ,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, m ∈N. [33]

Definition 3

The Riemann-Liouville fractional differential operator of order α ≥0, [1,30]

Dαft=dmdtmIm−αft,m−1<α≤m,m∈N.

Definition 4

The left sided caputo of f (t) derivative is defined as [1]

Dtαft=Jm−αDnft,1Γn−α∫0tt−Tm−α−1fmτdτ,

where m - 1 < α ≤ m, m ∈N, t > 0.

Definition 5

Ji–Huan He‘s fractional derivative is defined as [34]

∂αft∂tα=1Γn−α∂n∂tn∫0tτ−tn−α−1f0τ−fτdτ,n−1<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=ft∃M,τ1,τ2>0,ft<Metτj,ift∈−1j×0,∞,

by the following formula

f¯u=Sft=∫0∞fute−tdt,u∈−τ1,τ2.

Definition 7

The sumudu transform of f (t) = tα is defined as [35]

Stα=∫0∞e−ttαdt=Γα+1uα,Rα>0.

Definition 8

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

SIαft=u−αFu.

Definition 9

The Sumudu transform 𝕊[f (t)] of the Caputo fractional derivative is defined as [30,36]

SDtαft=u−αSft−∑k=0m−1u−α+kfk0+,

where

m−1<α≤m.

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,z∂x2βUx,y,z,t+gx,y,z∂y2βx,y,z,t+hx,y,z∂z2γx,y,z,t,0<x<a,0<y<0,0<z<c,0<β⩽1,0<γ⩽1.(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,tuα−∑k=0n−1Uk0uα−k=Sfx,y,z∂x2βUx,y,z,t+gx,y,z∂y2βUx,y,z,t+hx,y,z∂z2γUx,y,z,t(2)

Now, we define nonlinear operator as

Nφx,y,z,t;q=Sφx,y,z,t;q−∑k=0n−1Uk0u−k−uαSfx,y,z∂x2βφx,y,z,t;q+gx,y,z∂y2βφx,y,z,t;q+hx,y,z∂z2γφx,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:

1−qSφx,y,z,t;q−U0x,y,z,t=ℏqHx,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=1∞qmUmx,y,z,t(6)

where

Umx,y,z,t=1m_∂mφx,y,z,t;q∂qmq=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=1∞Umx,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,t−χmUm−1x,y,z,t=ℏHx,y,z,tRmU→m−1,x,y,z,t.(10)

Operating the inverse Sumudu transform on both sides, we get

Umx,t=χmUm−1x,y,z,t+ℏS−1Hx,y,z,tRmU→m−1,x,y,z,t,(11)

where

RmU→m−1,x,y,z,t=1m−1!∂m−1φx,y,z,t;q∂qm−1q=0(12)

and

χm=0,m≤1,1m>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=12x2∂x2βUx,t,0<x<1,0<α≤1,0<β≤1,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,t−S12x2∂x2βUx,t=0,
SDtαUx,t−∑k=0n−1Uk0u−k−uαS12x2∂x2βUx,t=0,

The nonlinear operator is defined by

Nφx,t;q=Sφx,t;q−∑k=0n−1φk0u−k−uαS12x2∂x2βφx,t;q,(15)

and thus

RmU→m−1,x,t=SUm−1x,t+uαS12x2∂x2βUm−1x,t.(16)

The mth- order deformation equation is given by

SUmx,t−χmUm−1x,t=ℏHx,tRmU→m−1,x,t.

Applying the inverse Sumudu transform, we have

Umx,t=χmUm−1x,t+S−1ℏHx,tRmU→m−1,x,t.(17)

Solving Eq. (17) by using Eq. (16) for m = 1, 2, ..., we obtain

U1x,t=−ℏx4−2βΓ4−2βtα+1Γα+2
U2x,t=−ℏx4−2βΓ4−2βtα+1Γα+2−ℏ2x4−2βΓ4−2βtα+1Γα+2+ℏ22Γ5−2βx4−2βΓ4−2βΓ5−4βt2α+1Γ2α+2
U3x,t=−ℏx4−2βΓ4−2βtα+1Γα+2−ℏ2x4−2βΓ4−2βtα+1Γα+2+ℏ22Γ5−2βx4−2βΓ4−2βΓ5−4βt2α+1Γ2α+2+ℏ−ℏx4−2βΓ4−2βtα+1Γα+2−ℏ2x4−2βΓ4−2βtα+1Γα+2+ℏ22Γ5−2βx4−2βΓ4−2βΓ5−4βt2α+1Γ2α+2ℏ22Γ5−2βx6−4βΓ4−2βΓ5−4βt2α+1Γ2α+2+ℏ32Γ5−2βx6−4βΓ4−2βΓ5−4βt2α+1Γ2α+2−ℏ32Γ5−2βΓ7−4βx6−4βΓ4−2βΓ5−4βΓ7−6βt3α+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=1∞Umx,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+3tαx4−2βΓα+1Γ3−2β+t2αx6−4βΓ5−2β2Γ2α+1Γ5−4βΓ3−2β+t3αx8−6βΓ7−2βΓ5−2β4Γ3α+1Γ7−6βΓ5−4βΓ3−2β+.....(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<α≤1,0<β≤1,0<Γ≤1,t>0,(21)

subject to the boundary conditions

U0,y,t=0,U2π,y,t=0,Ux,0,t=0,Ux,2π,t=0,
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.
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,t−S∂x2βUx,y,t+∂y2γUx,y,t=0,
SDtαUx,y,t−∑k=0n−1Uk0u−k−uαS∂x2βUx,y,t+∂y2γUx,y,t=0.

The nonlinear operator is defined by

Nφx,y,t;q=Sφx,y,t;q−∑k=0n−1φk0u−k−uαS∂x2βφx,y,t;q+∂y2γφx,y,t;q,(22)

and thus

RmU→m−1,x,y,t=SUm−1x,y,t+uαS∂x2βUm−1x,y,t+∂y2γUm−1x,y,t.(23)

The mth- order deformation equation is given by

SUmx,y,t−χmUm−1x,y,t=ℏHx,y,tRmU→m−1,x,y,t.

Applying the inverse Sumudu transform, we have

Umx,y,t=χmUm−1x,y,t+S−1ℏHx,y,tRmU→m−1,x,y,t.(24)

Solving Eq. (24) using Eq. (23) for m = 1, 2, ..., we obtain

U1x,y,t=−ℏ∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=0∞−1jy2i+1Γ2j+2+∑i=1∞−1ix2i+1Γ2i+2∑j=0∞−1jy2i+1−2γΓ2j+2−2γtαΓα+1,
U2x,y,t=−ℏ∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=0∞−1jy2j+1Γ2j+2+∑i=0∞−1ix2i+1Γ2i+2∑j=1∞−1jy2j+1−2γΓ2j+2−2γtαΓα+1−ℏ2∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=0∞−1jy2j+1Γ2j+2+∑i=0∞−1ix2i+1Γ2i+2∑j=1∞−1jy2j+1−2γΓ2j+2−2γtαΓα+1,+ℏ2∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1++ℏ2∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+1
U3x,y,t=−ℏ∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=0∞−1jy2j+1Γ2j+2+∑i=0∞−1ix2i+1Γ2i+2∑j=1∞−1jy2j+1−2γΓ2j+2−2γtαΓα+1−ℏ2∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=0∞−1jy2j+1Γ2j+2+∑i=0∞−1ix2i+1Γ2i+2∑j=1∞−1jy2j+1−2γΓ2j+2−2γtαΓα+1+ℏ2∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1+ℏ2∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+1+ℏℏ∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=0∞−1jy2j+1Γ2j+2+∑i=0∞−1ix2i+1Γ2i+2∑j=1∞−1jy2j+1−2γΓ2j+2−2γtαΓα+1+ℏ2∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1+ℏ2∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1+ℏ2∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+1+ℏ2∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1+ℏ3∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1−ℏ3∑i=3∞−1ix2i+1−6βΓ2i+2−6β∑j=0∞−1jy2j+1Γ2j+2
+∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=2∞−1jy2i+1−4γΓ2j+2−4γt3αΓ3α+1+ℏ2∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+1+ℏ3∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+1−ℏ3∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=2∞−1jy2i+1−4γΓ2j+2−4γ+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=2∞−1jy2j+1−4γΓ2j+2−4γ+∑i=0∞−1ix2i+1Γ2i+2∑j=3∞−1jy2i+1−6γΓ2j+2−6γt3αΓ3α+1+ℏ2∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1+ℏ2∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+1+ℏ2∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1+ℏ3∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1−ℏ3∑i=3∞−1ix2i+1−6βΓ2i+2−6β∑j=0∞−1jy2j+1Γ2j+2+∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=1∞−1jy2i+1−2γΓ2j+2−2γ+∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=2∞−1jy2i+1−4γΓ2j+2−4γt3αΓ3α+1
+ℏ2∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+1+ℏ3∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+1−ℏ3∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=2∞−1jy2i+1−4γΓ2j+2−4γ+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=2∞−1jy2j+1−4γΓ2j+2−4γ+∑i=0∞−1ix2i+1Γ2i+2∑j=3∞−1jy2i+1−6γΓ2j+2−6γt3αΓ3α+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=1∞Umx,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=1∞−1ix2i+1−2βΓ2i+2−2β∑j=0∞−1jy2i+1Γ2j+2+∑i=1∞−1ix2i+1Γ2i+2∑j=0∞−1jy2i+1−2γΓ2j+2−2γtαΓα+1+∑i=2∞−1ix2i+1−4βΓ2i+2−4β∑j=0∞−1jy2j+1Γ2j+2+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2i+1−2γΓ2j+2−2γt2αΓ2α+1+∑i=1∞−1ix2i+1−2βΓ2i+2−2β∑j=1∞−1jy2j+1−2γΓ2j+2−2γ+∑i=0∞−1ix2i+1Γ2i+2∑j=2∞−1jy2i+1−4γΓ2j+2−4γt2αΓ2α+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=e−2tsinxsiny(27)

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

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.
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+136x2∂x2βUx,y,z,t+y2∂y2γUx,y,z,t+z2∂z2δUx,y,z,t,0<x,y,z<1,0<α⩽1,0<β⩽1,0<γ⩽1,0<δ⩽1,t>0,(28)

subject to the boundary conditions

U0,y,z,t=0,U1,y,z,t=y4z4et−1,Ux,0,z,t=0,Ux,1,z,t=x4z4et−1,Ux,y,0,t=0,Ux,y,1,t=x4y4et−1

and the initial condition

Ux,y,z,0=0.

Applying the sumudu transform on both sides in Eq. (28),

SDtαUx,y,t−Sx4y4z4+136x2∂x2βUx,y,z,t+y2∂y2γUx,y,z,t+z2∂z2δUx,y,z,t=0,
SDtαUx,y,t−∑k=0n−1Uk0u−k−uαSx4y4z4+136x2∂x2βUx,y,z,t+y2∂y2γUx,y,z,t+z2∂z2δUx,y,z,t=0.

The nonlinear operator is defined by

Nφx,y,z,t;q=φx,y,z,t;q−∑k=0n−1φk0u−k−uαx4y4z4+136x2∂x2βφx,y,z,t;q+y2∂y2γφx,y,z,t;q+z2∂z2δUφx,y,z,t;q,(29)

and thus

RmU→m−1,x,y,z,t=SUm−1x,y,z,t+uαS(1−χm)x4y4z4+136x2∂x2βUx,y,z,t+y2∂y2γUx,y,z,t+z2∂z2δUx,y,z,t.(30)

The mth- order deformation equation is given by

SUmx,y,z,t−χmUm−1x,y,z,t=ℏHx,y,z,tRmU→m−1,x,y,z,t.

Applying the inverse Sumudu transform, we have

Umx,y,z,t=χmUm−1x,y,z,t+S−1ℏHx,y,z,tRmU→m−1,x,y,z,t.(31)

Solving Eq. (31) using Eq.(30) for m = 1, 2, ..., we get

U1x,y,z,t=−ℏx4y4z4tαΓα+1,
U2x,y,z,t=−ℏx4y4z4tαΓα+1−ℏ2x4y4z4tαΓα+1+ℏ236Γ5x6−2βy4z4Γ5−2β+Γ5x4y6−2Γz4Γ5−2γ+Γ5x4y4z6−2δΓ5−2δt2αΓ2α+1
U3x,y,z,t=−ℏx4y4z4tαΓα+1−ℏ2x4y4z4tαΓα+1+ℏ236Γ5x6−2βy4z4Γ5−2β+Γ5x4y6−2γz4Γ5−2γ+Γ5x4y4z6−2δΓ5−2δt2αΓ2α+1+ℏ−ℏx4y4z4tαΓα+1−ℏ2x4y4z4tαΓα+1+ℏ236Γ5x6−2βy4z4Γ5−2β+Γ5x4y6−2γz4Γ5−2γ+Γ5x4y4z6−2δΓ5−2δt2αΓ2α+1+ℏ236Γ5x6−2βy4z4Γ5−2βt2αΓ2α+1+ℏ336Γ5x6−2βy4z4Γ5−2βt2αΓ2α+1−ℏ236Γ5Γ7−2βx8−4βy4z4Γ7−4βΓ5−2β+Γ52x6−2βy6−2γz4Γ5−2βΓ5−2γ+Γ52x6−2βy4z6−2δΓ5−2βΓ5−2δt3αΓ3α+1+ℏ236Γ5x4y6−2γz4Γ5−2γt2αΓ2α+1+ℏ336Γ5x4y6−2γz4Γ5−2γt2αΓ2α+1−ℏ236Γ52x6−2βy6−2γz4Γ5−2βΓ5−2γ+Γ5Γ7−2γx4y8−4γz4Γ5−2γΓ7−4γ+Γ52x4y6−2γz6−2δΓ5−2γΓ5−2δt3αΓ3α+1+ℏ236Γ5x4y4z6−2δΓ5−2δt2αΓ2α+1+ℏ336Γ5x4y4z6−2δΓ5−2δt2αΓ2α+1−ℏ236Γ52x6−2βy4z6−2δΓ5−2βΓ5−2δ+Γ52x4y6−2γz6−2δΓ5−2γΓ5−2δ+Γ5Γ7−2δx4y8−4γz4Γ5−2δΓ7−4δt3αΓ3α+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=1∞Umx,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+2t2αx6−2βy4z43Γ1+2αΓ5−2β+t3αx8−4βy4z4Γ7−2β54Γ1+3αΓ7−4βΓ5−2β+2t2αx4y6−2γz43Γ1+2αΓ5−2γ+8t3αx6−2βy6−2γz49Γ1+3αΓ5−2γΓ5−2β+t3αx4y8−4γz4Γ7−2γ54Γ1+3αΓ7−4γΓ5−2γ+2t3αx4y4z6−2δ3Γ1+3αΓ5−2δ+4t3αx6−2βy4z6−2δ9Γ1+3αΓ5−2βΓ5−2δ+4t3αx4y6−2γz6−2δ9Γ1+3αΓ5−2βΓ5−2δ+4t3αx4y6−2γz6−2δ9Γ1+3αΓ5−2γΓ5−2δ+4t3αx4y6−2γz6−2δ9Γ1+3αΓ5−2γΓ5−2δ+t3αx4y4z8−4δΓ7−2δ54Γ1+3αΓ7−4δΓ5−2δ(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=x4y4z4et−1(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=12x2∂x2βUx,t,0<x<1,1<α≤2,0<β≤1,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,t−S12x2∂x2βUx,t=0,
SDtαUx,t−∑k=0n−1Uk0u−k−uαS12x2∂x2βUx,t=0.
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.
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;q−∑k=0n−1φk0u−k−uαS12x2∂x2βφx,t;q,(36)

and thus

RmU→m−1,x,t=SUm−1x,t+uαS12x2∂x2βUm−1x,t.(37)

The mth- order deformation equation is given by

SUmx,t−χmUm−1x,t=ℏHx,tRmU→m−1,x,t.

Applying the inverse Sumudu transform, we have

Umx,t=χmUm−1x,t+S−1ℏHx,tRmU→m−1,x,t.(38)

Solving Eq. (38) using Eq.(37) for m = 1, 2, ..., we get

U1x,t=−ℏx4−2βΓ4−2βtα+1Γα+2U2x,t=−ℏx4−2βΓ4−2βtα+1Γα+2−ℏ2x4−2βΓ4−2βtα+1Γα+2+ℏ22Γ5−2βx6−4βΓ4−2βΓ5−4βt2α+1Γ2α+2
U3x,t=−ℏx4−2βΓ4−2βtα+1Γα+2−ℏ2x4−2βΓ4−2βtα+1Γα+2+ℏ22Γ5−2βx4−2βΓ4−2βΓ5−4βt2α+1Γ2α+2+ℏ−ℏx4−2βΓ4−2βtα+1Γα+2−ℏ2x4−2βΓ4−2βtα+1Γα+2+ℏ22Γ5−2βx4−2βΓ4−2βΓ5−4βt2α+1Γ2α+2ℏ22Γ5−2βx6−4βΓ4−2βΓ5−4βt2α+1Γ2α+2+ℏ32Γ5−2βx6−4βΓ4−2βΓ5−4βt2α+1Γ2α+2−ℏ32Γ5−2βΓ7−4βx8−6βΓ4−2βΓ5−4βΓ7−6βt3α+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=1∞Umx,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+1x4−2βΓα+2Γ4−2β+t2α+1x6−4βΓ5−2β2Γ2α+2Γ5−4βΓ4−2β+t3α+1x8−6βΓ7−2βΓ5−2β4Γ3α+2Γ7−6βΓ5−4βΓ4−2β+.....(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.

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.
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+12x2∂x2βUx,y,z,t+y2∂y2γUx,y,z,t+z2∂z2δUx,y,z,t,0<x,y,z<1,1<α⩽2,0<β⩽1,0<Γ⩽1,0<δ⩽1,t>0,(42)

subject to the boundary conditions

0,y,z,t=y2et−1+z2e−t−1,U1,y,z,t=1+y2et−1+z2e−t−1,Ux,0,z,t=x2et−1+z2e−t−1,Ux,1,z,t=1+x2et−1+z2e−t−1,Ux,y,0,t=x2+y2et−1,Ux,y,1,t=x2+y2et−1+e−t−1

and the initial condition

Ux,y,z,0=0,Utx,y,z,0=x2+y2−z2.

Applying the sumudu transform on both sides in Eq. (42),

SDtαUx,y,t−Sx2+y2+z2+12x2∂x2βUx,y,z,t+y2∂y2γUx,y,z,t+z2∂z2δUx,y,z,t=0,
SDtαUx,y,t−∑k=0n−1Uk0u−k−uαSx2+y2+z2+12x2∂x2βUx,y,z,t+y2∂y2γUx,y,z,t+z2∂z2δUx,y,z,t=0.

The nonlinear operator is defined by

Nφx,y,z,t;q=Sφx,y,z,t;q−∑k=0n−1φk0u−k−uαSx2+y2+z2+12x2∂x2βφx,y,z,t;q+y2∂y2γφx,y,z,t;q+z2∂z2δφx,y,z,t;q,(43)

and thus

RmU→m−1,x,y,z,t=SUm−1x,y,z,t+uαS1−χmx2+y2+z2+12x2∂x2βUx,y,z,t+y2∂y2γUx,y,z,t+z2∂z2δUx,y,z,t(44)

The mth- order deformation equation is given by

SUmx,y,z,t−χmUm−1x,y,z,t=ℏHx,y,z,tRmU→m−1,x,y,z,t.

Applying the inverse Sumudu transform, we have

Umx,y,z,t=χmUm−1x,y,z,t+S−1ℏHx,y,z,tRmU→m−1,x,y,z,t.(45)

We start with

U0x,y,z,t=x2+y2−z2t+Jαx2+y2−z2t,=x2+y2t+tαΓα+1+z2−t+tαΓα+1

Solving Eq. (45) using Eq.(44) for m = 1, 2, ..., we get

U1x,y,z,t=x2+y2t+tαΓα+1+z2−t+tαΓα+1+ℏx2+y2t+tαΓα+1+z2−t+tαΓα+1+ℏx4−2βΓ3−2β+y4−2γΓ3−2γtα+1Γα+2+t2αΓ2α+1−z4−2δΓ3−2δ−tα+1Γα+2+t2αΓ2α+1,U2x,y,z,t=x2+y2t+tαΓα+1+z2−t+tαΓα+1+ℏx2+y2t+tαΓα+1+z2−t+tαΓα+1ℏx4−2βΓ3−2β+y4−2γΓ3−2γtα+1Γα+2+t2αΓ2α+1−z4−2δΓ3−2δ−tα+1Γα+2+t2αΓ2α+1+ℏx2+y2t+tαΓα+1+z2−t+tαΓα+1+ℏx2+y2t+tαΓα+1+z2−t+tαΓα+1ℏx4−2βΓ3−2β+y4−2γΓ3−2γtα+1Γα+2+t2αΓ2α+1−z4−2δΓ3−2δ−tα+1Γα+2+t2αΓ2α+1−ℏx4−2βΓ3−2βtα+1Γα+2+t2αΓ2α+1−ℏ2x4−2βΓ3−2βtα+1Γα+2+t2αΓ2α+1+ℏ22Γ5−2βx6−4βΓ3−2βΓ5−4βt2α+1Γ2α+2+t3αΓ3α+1−ℏy4−2γΓ3−2γtα+1Γα+2+t2αΓ2α+1−ℏ2y4−2γΓ3−2γtα+1Γα+2+t2αΓ2α+1+ℏ22Γ5−2γy6−4γΓ3−2γΓ5−4γt2α+1Γ2α+2+t3αΓ3α+1−ℏz4−2δΓ3−2δtα+1Γα+2+t2αΓ2α+1−ℏ2z4−2δΓ3−2δtα+1Γα+2+t2αΓ2α+1+ℏ22Γ5−2δz6−4δΓ3−2δΓ5−4δt2α+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=1∞Umx,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+2t2αx6−2βy4z43Γ1+2αΓ5−2β+t3αx8−4βy4z4Γ7−2β54Γ1+3αΓ7−4βΓ5−2β+2t2αx4y6−2γz43Γ1+2αΓ5−2γ+8t3αx6−2βy6−2γz49Γ1+3αΓ5−2γΓ5−2β+t3αx4y8−4γz4Γ7−2γ54Γ1+3αΓ7−4γΓ5−2γ+2t3αx4y4z6−2δ3Γ1+3αΓ5−2δ+4t3αx6−2βy4z6−2δ9Γ1+3αΓ5−2βΓ5−2δ+4t3αx4y6−2γz6−2δ9Γ1+3αΓ5−2βΓ5−2δ+4t3αx4y6−2γz6−2δ9Γ1+3αΓ5−2γΓ5−2δ+4t3αx4y6−2γz6−2δ9Γ1+3αΓ5−2γΓ5−2δ+t3αx4y4z8−4δΓ7−2δ54Γ1+3αΓ7−4δΓ5−2δ(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=x4y4z4et−1(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.

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.
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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Received: 2016-9-14
Accepted: 2016-12-15
Published Online: 2017-3-17

© 2017 Rishi Kumar Pandey and Hradyesh Kumar Mishra

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2017-0009
Keywords for this article
He‘s fractional derivative; Fractional partial differential equation; Heat equations; Homotopy Analysis Fractional Sumudu Transform Method; Wave equation
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  57. Regular Articles
  58. Charge conservation in a gravitational field in the scalar ether theory
  59. Regular Articles
  60. Measurement problem and local hidden variables with entangled photons
  61. Regular Articles
  62. Compression of hyper-spectral images using an accelerated nonnegative tensor decomposition
  63. Regular Articles
  64. Fabrication and application of coaxial polyvinyl alcohol/chitosan nanofiber membranes
  65. Regular Articles
  66. Calculating degree-based topological indices of dominating David derived networks
  67. Regular Articles
  68. The structure and conductivity of polyelectrolyte based on MEH-PPV and potassium iodide (KI) for dye-sensitized solar cells
  69. Regular Articles
  70. Chiral symmetry restoration and the critical end point in QCD
  71. Regular Articles
  72. Numerical solution for fractional Bratu’s initial value problem
  73. Regular Articles
  74. Structure and optical properties of TiO2 thin films deposited by ALD method
  75. Regular Articles
  76. Quadruple multi-wavelength conversion for access network scalability based on cross-phase modulation in an SOA-MZI
  77. Regular Articles
  78. Application of ANNs approach for wave-like and heat-like equations
  79. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  80. Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory
  81. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  82. A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
  83. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  84. Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
  85. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  86. On the libration collinear points in the restricted three – body problem
  87. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  88. Research on Critical Nodes Algorithm in Social Complex Networks
  89. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  90. A simulation based research on chance constrained programming in robust facility location problem
  91. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  92. A mathematical/physics carbon emission reduction strategy for building supply chain network based on carbon tax policy
  93. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  94. Mathematical analysis of the impact mechanism of information platform on agro-product supply chain and agro-product competitiveness
  95. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  96. A real negative selection algorithm with evolutionary preference for anomaly detection
  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  164. Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  166. The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  182. Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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