Modified Fractional Reduced Differential Transform Method for the Solution of Multiterm Time-Fractional Diffusion Equations

Abuasad, Salah; Hashim, Ishak; Karim, Samsul Ariffin Abdul

doi:10.1155/2019/5703916

Cited by 26 publications

(18 citation statements)

References 27 publications

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“…Hence, from the properties of one‐dimensional differential transform method, 18

\begin{matrix} lefttrue \\ ϕ (t, x_{1}, x_{2}, .., x_{n}) = ({false\sum}_{i_{1} = 0}^{\infty} ϕ_{1} (i_{1}) x_{1}^{i_{1}}) ({false\sum}_{i_{2} = 0}^{\infty} ϕ_{2} (i_{2}) x_{2}^{i_{2}}) . \dots ({false\sum}_{i_{n} = 0}^{\infty} ϕ_{n} (i_{n}) x_{n}^{i_{n}}) ({false\sum}_{j = 0}^{\infty} h (j) t^{j}) \\ = {false\sum}_{i_{1} = 0}^{\infty} {false\sum}_{i_{2} = 0}^{\infty} . \dots {false\sum}_{i_{n} = 0}^{\infty} {false\sum}_{j = 0}^{\infty} ϕ (i_{1} i_{2} \dots i_{n} j) x_{1}^{i_{1}} x_{2}^{i_{2}} \dots x_{n}^{i_{n}} t^{j}, \end{matrix}

where ϕ ( i 1 , i 2 , …, i n , j ) = ϕ 1 ( i 1 ) ϕ 2 ( i 2 )… ϕ n ( i n ) h ( j ) is denoted as the spectrum of ϕ ( t , x 1 , x 2 , .., x n ). Also, ϕ ( t , x 1 , x 2 , .., x n ) is called the original function and ϕ k ( x 1 , x 2 , .., x n ) is called the transformed function or T‐function.…”

Section: Fractional Reduced Differential Transform Methodsmentioning

confidence: 99%

“…A fractional-order integrodifferential equation with nonlocal boundary conditions has been solved in Nazari and Shahmorad 13 using the FRDTM to convert this equation into a system of algebraic equations and solve it to get the unknowns. In addition, many other fractional models have been solved using this method including fractional differential-algebraic equations, 14 time-fractional vibration model, 15 fractional Riccati equation, 16,17 multiterm time-fractional diffusion equations, 18 irrational-order fractional equations, 19 Bagley-Torvik equation, 20 fuzzy fractional dynamical model of marriage, 21 system of fractional differential equations, 22 and nonlinear fractional Klein-Gordon equation. 23 Due to the importance of PDE in the modeling of various scientific fields, several methods have been proposed for their solution.…”

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confidence: 99%

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On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

Jena

Sedighi

2020

Math Methods in App Sciences

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A widescale description of various phenomena in science and engineering such as physics, chemical, acoustics, control theory, finance, economics, mechanical engineering, civil engineering, and social sciences is well described by nonlinear fractional differential equations (NLFDEs). In turbulence, fluid dynamics, and nonlinear biological systems, applications of NLFDEs can also be found. NLFDEs are believed to be powerful tools to describe real-world problems more precisely than the differential equation of the integer-order. In this research, we have used the fractional reduced differential transform method (FRDTM) to find the solution of the time-fractional Sawada-Kotera-Ito seventh-order equation. The novelty of the FRDTM is that it does not require any discretization, transformation, perturbation, or any restrictive conditions. In addition, compared to other methods, this approach needs less calculation. For special cases of an integer and noninteger orders, computed results are compared with existing results. Present results are in good agreement with the existing solutions. Here, the fractional derivatives are considered in the Caputo sense. Convergence analysis of the results has also been studied with the increasing number of terms of the solution.

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“…Hence, from the properties of one‐dimensional differential transform method, 18

\begin{matrix} lefttrue \\ ϕ (t, x_{1}, x_{2}, .., x_{n}) = ({false\sum}_{i_{1} = 0}^{\infty} ϕ_{1} (i_{1}) x_{1}^{i_{1}}) ({false\sum}_{i_{2} = 0}^{\infty} ϕ_{2} (i_{2}) x_{2}^{i_{2}}) . \dots ({false\sum}_{i_{n} = 0}^{\infty} ϕ_{n} (i_{n}) x_{n}^{i_{n}}) ({false\sum}_{j = 0}^{\infty} h (j) t^{j}) \\ = {false\sum}_{i_{1} = 0}^{\infty} {false\sum}_{i_{2} = 0}^{\infty} . \dots {false\sum}_{i_{n} = 0}^{\infty} {false\sum}_{j = 0}^{\infty} ϕ (i_{1} i_{2} \dots i_{n} j) x_{1}^{i_{1}} x_{2}^{i_{2}} \dots x_{n}^{i_{n}} t^{j}, \end{matrix}

Section: Fractional Reduced Differential Transform Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

Jena

Sedighi

2020

Math Methods in App Sciences

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“…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][31][32][33].…”

Section: Fractional Calculusmentioning

confidence: 99%

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

Mussa

Gizaw

Negassa

2021

Mathematical Problems in Engineering

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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 when α = 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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“…Finite element method (FEM) is used to solve symmetric space-FPDEs with the Riesz fractional operator [15]. Besides these methods, some mathematicians have used other methods such as Jacobi elliptic expansion method [16], exp-function method [17], fractional reduced differential transform method (FRDTM) [18], Lie algebra method [19], finite difference method (FDM) [20], ( G G )-expansion method [21], [22] and many other numerical and analytical methods.…”

Section: And Baleanu Have Reviewed the Adm Besides Its Modifications mentioning

confidence: 99%

Families of Travelling Waves Solutions for Fractional-Order Extended Shallow Water Wave Equations, Using an Innovative Analytical Method

et al. 2019

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In the present research article, an efficient analytical technique is applied for travelling waves solutions of fractional partial differential equations. The investigated problems are reduced to ordinary differential equations, by a variable transformation. The solutions of the resultant ordinary differential equations are expressed in the term of some suitable polynomials, which provide trigonometric, hyperbolic and rational function solutions with some free parameters. To confirm the reliability and novelty of the current work, the proposed method is applied for the solutions of (2+1) and (3+1)-dimensional fractionalorder extended shallow water wave equations.

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Modified Fractional Reduced Differential Transform Method for the Solution of Multiterm Time-Fractional Diffusion Equations

Cited by 26 publications

References 27 publications

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

On the wave solutions of time‐fractional Sawada‐Kotera‐Ito equation arising in shallow water

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

Families of Travelling Waves Solutions for Fractional-Order Extended Shallow Water Wave Equations, Using an Innovative Analytical Method

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