scholarly journals Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Guodong Shi ◽  
Yanlei Gong ◽  
Mingxu Yi
Keyword(s):  
Differential Equations ◽  
Legendre Polynomials ◽  
Numerical Scheme ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Tabular Form ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Singular Kernels

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

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.

Haar wavelet operational matrix method for system of fractional nonlinear differential equations

2017 ◽  
Vol 15 (05) ◽  
pp. 1750043 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Reliable Method ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Implementation Process ◽  
Haar Wavelet ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

scholarly journals Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

2021 ◽  
Vol 5 (3) ◽  
pp. 70
Author(s):  
Esmail Bargamadi ◽  
Leila Torkzadeh ◽  
Kazem Nouri ◽  
Amin Jajarmi
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Algebraic System ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Chebyshev Wavelets ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Chebyshev Wavelet

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

scholarly journals An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1755
Author(s):  
M. S. Al-Sharif ◽  
A. I. Ahmed ◽  
M. S. Salim
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Fractional Differential

Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

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.

Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel

2021 ◽  
Vol 477 (2253) ◽  
Author(s):  
Chandrali Baishya ◽  
P. Veeresha
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Laguerre Polynomial ◽  
Fractional Integration ◽  
Laguerre Polynomials ◽  
Operational Matrix ◽  
Computational Technique ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Differential

The Atangana–Baleanu derivative and the Laguerre polynomial are used in this analysis to define a new computational technique for solving fractional differential equations. To serve this purpose, we have derived the operational matrices of fractional integration and fractional integro-differentiation via Laguerre polynomials. Using the derived operational matrices and collocation points, we reduce the fractional differential equations to a system of linear or nonlinear algebraic equations. For the error of the operational matrix of the fractional integration, an error bound is derived. To illustrate the accuracy and the reliability of the projected algorithm, numerical simulation is presented, and the nature of attained results is captured in diverse order. Finally, the achieved consequences enlighten that the solutions obtained by the proposed scheme give better convergence to the actual solution than the results available in the literature.

A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Zain ul Abdeen ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Haar Wavelets ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Content Type ◽  
Fractional Differential

PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.Design/methodology/approachThe aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.FindingsThe upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.Originality/valueThe numerical method is purposed for solving Hadamard-type fractional differential equations.

scholarly journals Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Mikhail Bulatov ◽  
Pedro Lima ◽  
Ewa Weinmüller
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Solution ◽  
Main Stream ◽  
Algebraic Equations ◽  
Uniqueness Of Solutions ◽  
The Third ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Dimensional Integral

AbstractWe consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.

Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation

2019 ◽  
Vol 17 (04) ◽  
pp. 1950026 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Type Equation ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential ◽  
Quasilinearization Technique ◽  
Distribution Equation

In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

Sign in / Sign up

Export Citation Format

Share Document