New operational matrices of orthogonal Legendre polynomials and their operational

Abstract

Many conventional physical and engineering phenomena have been identified to be well expressed by making use of the fractional order partial differential equations (FOPDEs). For that reason, a proficient and stable numerical method is needed to find the approximate solution of FOPDEs. This article is designed to develop the numerical scheme able to find the approximate solution of generalized fractional order coupled systems (FOCSs) with mixed partial derivative terms of fractional order. Our main objective in this article is the development of a new operational matrix for fractional mixed partial derivatives based on the orthogonal shifted Legendre polynomials (SLPs). The fractional derivatives are considered herein in the sense of Caputo. The proposed method has the advantage to reduce the considered problems to a system of algebraic equations which are simple in handling by any computational software. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some examples are included to demonstrate the accuracy and validity of the proposed method. ARTICLE HISTORY

A novel fractal-fractional analysis of the stellar helium burning network using extended operational matrix method

“…Because the majority of FDEs are complicated to solve analytically, numerical solutions are in high demand. A variety of numerical approaches are present in the literature for solving ordinary and partial FDEs numerically, but the OM approach coupled with the Tau method and the collocation method is commonly utilized [21][22][23][24][25][26][27][28][29][30]. This method works by converting the FDEs into an algebraic equation system that can be solved with any computer programmer.…”

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

“…Many engineering, physics, and other scientific phenomena can be accurately modeled using fractional numerical methods, such as the theory of arbitrary-order derivatives and integrals. [1][2][3][4][5][6] Theoretical and practical developments in fractional calculus have occurred recently. In almost all applied sciences, fractional calculus has been used to explain numerous phenomena.…”

“…Fractional calculus has advanced significantly in theory and application over the last few decades. Many engineering, physics, and other scientific phenomena can be accurately modeled using fractional numerical methods, such as the theory of arbitrary‐order derivatives and integrals 1‐6 . Theoretical and practical developments in fractional calculus have occurred recently.…”

Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials

“…Additionally, Bharway et al [25] introduced a new shifted Chebyshev operational matrix of fractional integration for solving linear FDEs. Moreover, Talib et al [26] developed a new operational matrix based on the orthogonal shifted Legendre polynomials to numerically solve the fractional partial differential equations. Meanwhile, Rahimkhani et al [27] introduced a Bernoulli wavelet operational matrix of fractional integration for obtaining the approximate solution of a fractional delay differential equation.…”