scholarly journals Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel

2021 ◽  
Vol 6 (1) ◽  
pp. 19
Author(s):  
Mohamed A. Abdelkawy ◽  
Ahmed Z. M. Amin ◽  
António M. Lopes ◽  
Ishak Hashim ◽  
Mohammed M. Babatin
Keyword(s):  
Collocation Method ◽  
Fractional Order ◽  
Jacobi Polynomials ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Singular Kernel ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular

We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.

scholarly journals Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khadijeh Sadri ◽  
Kamyar Hosseini ◽  
Dumitru Baleanu ◽  
Ali Ahmadian ◽  
Soheil Salahshour
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Polynomials ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Linear Algebraic Equations

AbstractThe shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

scholarly journals NUMERICAL SOLUTION OF VARIABLE-ORDER TIME FRACTIONAL WEAKLY SINGULAR PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS WITH ERROR ESTIMATION

2020 ◽  
Vol 25 (4) ◽  
pp. 680-701
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani ◽  
Mohsen Razzaghi
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Collocation Method ◽  
Error Estimation ◽  
Operational Matrices ◽  
Variable Order ◽  
Algebraic Equations ◽  
Laguerre Functions ◽  
Weakly Singular Kernel ◽  
Weakly Singular

In this paper, we apply Legendre-Laguerre functions (LLFs) and collocation method to obtain the approximate solution of variable-order time-fractional partial integro-differential equations (VO-TF-PIDEs) with the weakly singular kernel. For this purpose, we derive the pseudo-operational matrices with the use of the transformation matrix. The collocation method and pseudo-operational matrices transfer the problem to a system of algebraic equations. Also, the error analysis of the proposed method is given. We consider several examples to illustrate the proposed method is accurate.

scholarly journals A Numerical Solution of Hereditary Equations with a Weakly Singular Kernel for Vibration Analysis of Viscoelastic Systems / Vienâdojumu Ar Vâjo Singulâro Kodolu Skaitliskais Risinâjums Iedzimto Viskoelastîgo Sistçmu Vibrâciju Analîzei

2015 ◽  
Vol 69 (6) ◽  
pp. 326-330 ◽  
Author(s):  
Botir Usmonov
Keyword(s):  
Singular Kernel ◽  
Dynamic Loads ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Exponential Kernel ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Numerical Formulation ◽  
The Galerkin Method ◽  
Deformable System

Abstract Viscoelastic, or composite materials that are hereditary deformable, have been characterised by exponential and weakly singular kernels in a hereditary equation. An exponential kernel is easy to be numerically implemented, but does not well describe complex vibratory behaviour of a hereditary deformable system. On the other hand, a weakly singular kernel is known to describe the complex vibratory behaviour, but is nontrivial to be numerically implemented. This study presents a numerical formulation for solving a hereditary equation with a weakly singular kernel. Recursive algebraic equations, which are numerically solvable, are formulated by using the Galerkin method enhanced by a numerical integration and elimination of weak singularity. Numerical experiments showed that the present approach with a weakly singular kernel is well fitted into a realistic vibratory behaviour of a hereditary deformable system under dynamic loads, as compared to the same approach with an exponential kernel.

A collocation method for numerical solutions of fractional-order logistic population model

2016 ◽  
Vol 09 (02) ◽  
pp. 1650031 ◽  
Author(s):  
Şuayip Yüzbaşı
Keyword(s):  
Approximate Solution ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Fractional Order ◽  
Population Model ◽  
Numerical Solutions ◽  
Model Problem ◽  
Error Function ◽  
Approximate Solutions ◽  
Algebraic Equations

In this study, a collocation technique is presented for approximate solution of the fractional-order logistic population model. Actually, we develop the Bessel collocation method by using the fractional derivative in the Caputo sense to obtain the approximate solutions of this model problem. By means of the fractional derivative in the Caputo sense, the collocation points, the Bessel functions of the first kind, the method transforms the model problem into a system of nonlinear algebraic equations. Numerical applications are given to demonstrate efficiency and accuracy of the method. In applications, the reliability of the scheme is shown by the error function based on the accuracy of the approximate solution.

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