Numerical Multistep Approach for Solving Fractional Partial Differential Equations

Abstract

In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep reduced differential transformation method (MRDTM), is a simple amendment of the reduced differential transformation method, in which it is treated as an algorithm in a sequence of small intervals, in order to hold out accurate approximate solutions over a longer time frame compared to the traditional RDTM. The fractional derivatives are described in the Caputo sense, while the behavior of solutions for different values of fractional order [Formula: see text] compared with exact solutions is shown graphically. The analysis is accompanied by four test examples to demonstrate that the proposed approach is reliable, fully compatible with the complexity of these equations, and can be strongly employed for many other nonlinear problems in fractional calculus.

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

“…In the last decades, the topic of fractional calculus has attracted the attention of numerous researchers for its considerable importance in many applications such as fluid dynamics, viscoelasticity, physics, entropy theory and vibrations [10][11][12][13][14]. In this regard, many differential equations of integer order were generalized to fractional order, as well as various methods were developed to solve them.…”

A computational modelling of micro strip patch antenna and its solution by RDTM

“…3 and 5 show the plots of Eqs. (18) and (20) and the numerical results of Example 4.1 and Example 4.2 through various graphs. The variation of voltage vðx; tÞ with distance x at a particular time t=0.5 and t = 1.0 has been shown in Fig.…”

“…multistep reduced differential transform method [19,20], reproducing kernel Hilbert space method [21,22] and multistep generalized differential transform method [23] are also capable to handle a wide range of problems. The multistep reduced differential transform method is proposed by Al-Smadi et al [19,20].…”

“…multistep reduced differential transform method [19,20], reproducing kernel Hilbert space method [21,22] and multistep generalized differential transform method [23] are also capable to handle a wide range of problems. The multistep reduced differential transform method is proposed by Al-Smadi et al [19,20]. AlSmadi et al [21] applied reproducing kernel Hilbert space method to approximate the solution two-point boundary value problems for fourth-order Fredholm-Volterra integrodifferential equations whereas the authors of [22] applied aforesaid method to obtain the solution for systems of second-order differential equations with periodic boundary conditions.…”

Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

“…Therefore, it is necessary to obtain some mathematical tools to understand the complex structure of uncertainty models [1][2][3][4][5]. On the other hand, the theory of fractional calculus, which is a generalization of classical calculus, deals with the discussion of the integrals and derivatives of noninteger order, has a long history, and dates back to the seventeenth century [6][7][8][9][10]. Different forms of fractional operators are introduced to study FDEs such as Riemann-Liouville, Grunwald-Letnikov, and Caputo.…”