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In this paper, Daftardar-Gejji and Jafari method is applied to solve fractional heat-like and wave-like models with variable coefficients. The method is proved for a variety of problems in one, two and three dimensional spaces where analytical approximate solutions are obtained. The examples are presented to show the efficiency and simplicity of this method.

Various phenomena in engineering physics, chemistry, other sciences can be described very successfully by models using mathematical tools from fractional calculus, i.e., the theory of derivatives and integrals of fractional non-integer order [

The fractional calculus has a three-centuary and two decades long history. The idea appeared in a letter by Leibniz to Lâ€™Hôpital in 1695. The subject of fractional calculus has gained importance during the past three decades due mainly to its demonstrated applications in different area of physics and engineering. Indeed it provides several potentially useful tools for solving differential and integral equations. It is important to solve time fractional partial differential equations. It was found that fractional time derivatives arise generally as infinitesimal generators of the time evolution when taking a long-time scaling limit. Hence, the importance of investigating fractional equations arises from the necessity to sharpen the concepts of equilibrium, stability states, and time evolution in the long- time limit (for more details, see [

In this work, we will consider the fractional heat-like and wave-like equations of the form

subject to the Neumann boundary conditions

and the initial conditions

where

In the case of

Recently, Molliq et al. [

The Daftardar-Jafari method (DJM) developed in 2006 has been extensively used by many researchers for the treatment of linear and nonlinear ordinary and partial differential equations of integer and fractional order [

The main objective of this paper is to apply DJM to obtain fractional solutions for different models of Equation (1). While the VIM [

In this section we will introduce some definitions and properties of the fractional calculus to enable us to follow the solutions of the problem given in this paper. These definitions include, Riemann-Liouville, Weyl, Reize, Compos, Caputo, and Nashimoto fractional operators.

Definition 1. Let

for

Properties of the operator

For

1.

2.

3.

4.

The Riemann-Liouville derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we shall introduce now a modified fractional differential operator

Definition 2. The fractional derivative of

for

Also, we need here two of its basic properties.

Lemma 1. If

and

The Caputo fractional derivative is considered here because it allows traditional initial and boundary conditions to be included in the formulation of the problem. In this paper, we consider the three-dimensional time fractional heat- like and wave-like Equation (1.1), where the unknown function

Definition 3. For

For mathematical properties of fractional derivatives and integrals one can consult the above mentioned references.

Consider the following general functional equation

where

The nonlinear operator

From Equations (5) and (6), Equation (4) is equivalent to

We define the recurrence relation

Then

and

The

In this section, we illustrate our analysis by examining the following three frac- tional heat-like equations.

Example 1. Firstly, let us consider the one-dimensional initial boundary value problems (IBVP)

subject to the boundary conditions

and the initial condition

Operating with

For

Following the algorithm (8), the successive approximations are

Thus, the approximate solution in a series form is given by

So, the solution for the standard heat-like equation (

This series has the closed form

which is the exact solution of the problem (11)-(13) compatible with VIM, ADM and HAM.

Example 2. Now, let us consider the two-dimensional IBVP

subject to the Neumann boundary conditions

and the initial condition

Operating with

For

Using the algorithm (8), the successive approximations are

Thus, the approximate solution in a series form is given by

So, the solution for the standard heat-like equation (

This series has the closed form

which is the exact solution of the problem (14)-(16) compatible with VIM, ADM and HAM.

Example 3. Let us consider the three-dimensional inhomogeneous IBVP

subject to the boundary conditions

and the initial condition

Operating with

For

Utilizing the algorithm (8), the successive approximations are

Thus, the approximate solution in a series form is given by

So, the solution for the standard heat-like equation (

This series has the closed form

which is the exact solution of the problem (17)-(19) compatible with VIM, ADM and HAM.

In this section, we illustrate our analysis by examining the following three fractional wave-like equations.

Example 1. We first consider the one-dimensional IBVP

subject to the boundary conditions

and the initial conditions

Operating with

For

Following the algorithm (8), the successive approximations are

Thus, the approximate solution in a series form is given by

So, the solution for the standard heat-like equation (

This series has the closed form

which is the exact solution of the problem (20)-(22) compatible with VIM, ADM and HAM.

Example 2. We next consider the two-dimensional IBVP

subject to the Neumann boundary conditions

and the initial conditions

Operating with

For

Using the algorithm (8), the successive approximations are

Thus, the approximate solution in a series form is given by

So, the solution for the standard heat-like equation (

This series has the closed form

which is the exact solution of the problem (23)-(25) compatible with VIM, ADM and HAM.

Example 3. Finally, we consider the three-dimensional inhomogeneous IBVP

subject to the boundary conditions

and the initial conditions

Operating with

For

Utilizing the algorithm (8), the successive approximations are

So, the solution for the standard heat-like equation (

This series has the closed form

which is the exact solution of the problem (26)-(28) compatible with VIM, ADM and HAM.

In this work, DJM has been successfully used to solve fractional heat-like and wave-like equations with variable coefficients giving it a wider applicability. The proposed scheme was applied directly without any need for transformation formulae or restrictive assumptions. Results have shown that the analytical approximate solution process of DJM is compatible with those methods in the literature providing analytical approximation such as VIM, ADM and HAM. The results obtained in all studied cases demonstrate the reliability and the efficiency of this method.

Al-Hayani, W. (2017) Daftardar-Jafari Method for Fractional Heat-Like and Wave-Like Equations with Variable Coefficients. Applied Mathe- matics, 8, 215-228. https://doi.org/10.4236/am.2017.82018