Shifted Gegenbauer Operational Matrix and Its Applications for Solving Fractional Differential Equations

Melad

2022

Math Sci

The present study is related to the numerical solutions of new mathematical models based on the variable order Emden-Fowler delay differential equations. The shifted fractional Gegenbauer, $$C_{S,j}^{(\alpha ,\mu )}(t),$$ C S , j ( α , μ ) ( t ) , operational matrices (OMs) of VO differentiation, in conjunction with the spectral collocation method are used to solve aforementioned models numerically. The VO operator of differentiation will be used in the Caputo sense. The proposed technique simplifies these models by reducing them to systems of algebraic equations that are easy to solve. To determine the effectiveness and accuracy of the sugested technique, the absolute errors and maximum absolute errors for four realistic models are studied and illustrated by several tables and graphs at different values of the VO and the SFG parameters; $$\alpha$$ α and $$\mu.$$ μ . Also numerical comparisons between the suggested technique with other numerical methods in the existing literature are held. The numerical results confirm that the suggested technique is accurate, computationally efficient and easy to implement.

Section: Gegenbauer Polynomialsmentioning

confidence: 99%

“…• For = 1, the SFGPS reduced to the SGPs. Numerous studies have shown that the Gegenbauer polynomials (GPs) are very effective in solving a wide range of issues [32][33][34]. • The parameter 𝛼 > −0.5 distributes the Gegenbauer polynomials (GPs).…”

Section: Introductionmentioning

confidence: 99%

A new numerical strategy for solving nonlinear singular Emden-Fowler delay differential models with variable order

Melad

2022

Math Sci

“…The approximation methods based on spectral techniques and orthogonal functions have become a rigorous tool for solving various fractional dynamical systems [36,37]. These nonlocal methods are efficient numerical schemes for discretising the nonlocal fractional differential operators [38,39].…”

Section: Introductionmentioning

confidence: 99%

Improved Gegenbauer spectral tau algorithms for distributed-order time-fractional telegraph models in multi-dimensions

Hashem

2023

Numer Algor

The distributed-order fractional telegraph models are commonly used to describe the phenomenas of diffusion and wave-like anomalous, which can model processes without a power-law scale across the entire temporal domain. To increase the range of implementation of distributed-order fractional telegraph models, there is a need to present effective and accurate numerical algorithms to solve these models, as these models are hard to solve analytically. In this work, a novel matrix representation of the distributed-order fractional derivative based on shifted Gegenbauer (SG) polynomials has been derived. Also, two efficient algorithms based on the aforementioned operatonal matrix and the spectral tau method have been constructed for solving the one- and two-dimensional (1D and 2D) distributed-order time-fractional telegraph models with spatial variable coefficients. Also, a new operational matrix of the multiplication of space vectors has been built to have the ability in applying the tau method in the 2D case. The convergence and error bound analysis of the presented techniques are investigated. Moreover, the presented algorithms are applied on four miscellaneous test examples to illustrate the robustness and effectiveness of these algorithms.

“…The primary goal of this research is to extract an accurate numerical method to solve the following models: Spectral techniques are one of the most effective and powerful numerical algorithms for solving differential and integral equations with integer and non-integer order derivatives. Spectral approaches such as pseudo spectral [4], Galerkin [5], and Tau [17] can be produced from a weighted residual method. The spectral collocation method is becoming more widely used, with applications in a variety of fields.…”

Section: Introductionmentioning

confidence: 99%

“…It is characterized by exponential rates of convergence, great precision, ease of use, and effectiveness in handling a variety of problems. Besides the spectral methods, operational matrices are commonly utilized for solving these types of equations [17,5]. The current paper is based on the shifted fractional Gegenbauer polynomials for building the variable-order operational matrix of derivatives.…”

Section: Introductionmentioning

confidence: 99%

A New Numerical Mechanism for Solving Two Models of Variable Order Delay Diﬀerential Equations

Melad

Delta Journal of Science

Mahmoud

2022

Variable order delay differential equations with initial functions; Variable order Volterra delay integrodifferential equations; Operational matrix; Shifted fractional Gegenbauer polynomials; The method of steps; Collocation methodThe current paper offers an effective numerical mechanism for solving two models of variable order (VO) linear/nonlinear delay differential equations; the models represent the variable order delay differential equations (VODDEs) and the variable order Volterra delay integro-differential equations (VO-VDIDES) with initial functions. In the proposed mechanism, the method of steps is used to transfer the VO delay models (VO-DMs) into variable order non delay ones with initial conditions. After a novel operational matrix (OM) of VO derivative of the shifted fractional Gegenbauer polynomials (SFGPs) injunction with the spectral collocation method are utilized to transfer the aforementioned problem into a system of algebraic equations, which is simple to solve. The error estimation of the proposed technique is established through the article. The efficiency and accuracy of the proposed technique are verified by applying it to several numerical delay differential equations with constant or variable delay. The obtained numerical results are compared with other published data in the existing literature to expose the accuracy and efficiency of the proposed technique.