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2 Journal of Applied Mathematics and Computational Mechanics

2 2020

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A new numerical technique for solving fractional Bratu’s initial value problems in the Caputo and Caputo-Fabrizio sense

Khalouta Ali, Kadem Abdelouahab

Journal of Applied Mathematics and Computational Mechanics

2020

Vol. 19, nr 1

43--56

The purpose of this paper is to propose a new numerical technique called the natural decomposition method (NDM) for solving fractional Bratu’s initial value problems (FBIVP) in the Caputo and Caputo-Fabrizio sense. The NDM is a combined form of the natural transform method and the Adomian decomposition method. The numerical example is provided in order to validate the efficiency and reliability of the proposed method. The obtained results reveal that the proposed method is a very efficient and simple tool for solving fractional differential equations.

A new modification of the reduced differential transform method for nonlinear fractional partial differential equations

Khalouta Ali, Kadem Abdelouahab

Journal of Applied Mathematics and Computational Mechanics

2020

Vol. 19, nr 3

45--58

The objective of this study is to present a new modification of the reduced differential transform method (MRDTM) to find an approximate analytical solution of a certain class of nonlinear fractional partial differential equations in particular, nonlinear time-fractional wave-like equations with variable coefficients. This method is a combination of two different methods: the Shehu transform method and the reduced differential transform method. The advantage of the MRDTM is to find the solution without discretization, linearization or restrictive assumptions. Three different examples are presented to demonstrate the applicability and effectiveness of the MRDTM. The numerical results show that the proposed modification is very effective and simple for solving nonlinear fractional partial differential equations.