In this paper, we introduce a new type of integral transforms, called the ARA integral transform that is defined as: G n [ g ( t ) ] ( s ) = G ( n , s ) = s ∫ 0 ∞ t n − 1 e − s t g ( t ) d t , s > 0 . We prove some properties of ARA transform and give some examples. Also, some applications of the ARA transform are given.
In this article, a new, attractive method is used to solve fractional neutral pantograph equations (FNPEs). The proposed method, the ARA-Residual Power Series Method (ARA-RPSM), is a combination of the ARA transform and the residual power series method and is implemented to construct series solutions for dispersive fractional differential equations. The convergence analysis of the new method is proven and shown theoretically. To validate the simplicity and applicability of this method, we introduce some examples. For measuring the accuracy of the method, we make a comparison with other methods, such as the Runge–Kutta, Chebyshev polynomial, and variational iterative methods. Finally, the numerical results are demonstrated graphically.
In this paper, we use the ARA transform to solve families of fractional differential equations. New formulas about the ARA transform are presented and implemented in solving some applications. New results related to the ARA integral transform of the Riemann-Liouville fractional integral and the Caputo fractional derivative are obtained and the last one is implemented to create series solutions for the target equations. The procedure proposed in this article is mainly based on some theorems of particular solutions and the expansion coefficients of binomial series. In order to achieve the accuracy and simplicity of the new method, some numerical examples are considered and solved. We obtain the solutions of some families of fractional differential equations in a series form and we show how these solutions lead to some important results that include generalizations of some classical methods.
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