A New Bernoulli Wavelet Method for Numerical Solutions of Nonlinear Weakly Singular Volterra Integro-Differential Equations

2017 ◽  
Vol 14 (03) ◽  
pp. 1750022 ◽  
Author(s):  
P. K. Sahu ◽  
S. Saha Ray
Keyword(s):  
Differential Equations ◽  
Convergence Analysis ◽  
Present Method ◽  
Numerical Solutions ◽  
Bernoulli Polynomials ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Weakly Singular ◽  
Nonlinear Algebraic Equations ◽  
Bernoulli Wavelet

In this paper, Bernoulli wavelet method has been developed to solve nonlinear weakly singular Volterra integro-differential equations. Bernoulli wavelets are generated by dilation and translation of Bernoulli polynomials. The properties of Bernoulli wavelets and Bernoulli polynomials are first presented. The present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by Newton’s method. Convergence analysis of the present method has been discussed in this paper. Some illustrative examples have been demonstrated to show the applicability and accuracy of the present method.

Chebyshev wavelet method for numerical solutions of integro-differential form of Lane–Emden type differential equations

2017 ◽  
Vol 15 (02) ◽  
pp. 1750015 ◽  
Author(s):  
P. K. Sahu ◽  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Differential Form ◽  
Present Method ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Singular Differential Equation ◽  
Wavelet Method ◽  
Chebyshev Wavelets ◽  
Chebyshev Wavelet

In this paper, Chebyshev wavelet method (CWM) has been applied to solve the second-order singular differential equations of Lane–Emden type. Firstly, the singular differential equation has been converted to Volterra integro-differential equation and then solved by the CWM. The properties of Chebyshev wavelets were first presented. The properties of Chebyshev wavelets via Gauss–Legendre rule were used to reduce the integral equations to a system of algebraic equations which can be solved numerically by Newton’s method. Convergence analysis of CWM has been discussed. Illustrative examples have been provided to demonstrate the validity and applicability of the present method.

scholarly journals Solving Fuzzy Volterra Integrodifferential Equations of Fractional Order by Bernoulli Wavelet Method

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
R. Mastani Shabestari ◽  
R. Ezzati ◽  
T. Allahviranloo
Keyword(s):  
Integrodifferential Equation ◽  
Matrix Equation ◽  
Matrix Method ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Bernoulli Wavelet ◽  
Fractional Integrodifferential Equation

A matrix method called the Bernoulli wavelet method is presented for numerically solving the fuzzy fractional integrodifferential equations. Using the collocation points, this method transforms the fuzzy fractional integrodifferential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown coefficients. To illustrate the method, it is applied to certain fuzzy fractional integrodifferential equations, and the results are compared.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

2018 ◽  
Vol 9 (1-2) ◽  
pp. 16-27 ◽  
Author(s):  
Mohamed Abdel- Latif Ramadan ◽  
Mohamed R. Ali
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Bernoulli Wavelet

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

scholarly journals Bernoulli polynomials collocation for weakly singular Volterra integro-differential equations of fractional order

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3623-3635 ◽  
Author(s):  
Haman Azodi ◽  
Mohammad Yaghouti
Keyword(s):  
Differential Equations ◽  
Numerical Procedure ◽  
Bernoulli Polynomials ◽  
Bernoulli Polynomial ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Weakly Singular ◽  
The Matrix ◽  
Singular Kernels

This paper is concerned with a numerical procedure for fractional Volterra integro-differential equations with weakly singular kernels. The fractional derivative is in the Caputo sense. In this study, Bernoulli polynomial of first kind is used and its matrix form is given. Then, the matrix form based on the collocation points is constructed for each term of the problem. Hence, the proposed scheme simplifies the problem to a system of algebraic equations. Error analysis is also investigated. Numerical examples are announced to demonstrate the validity of the method.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

2019 ◽  
Vol 28 (14) ◽  
pp. 1950247 ◽  
Author(s):  
Sadiye Nergis Tural-Polat
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Riccati Differential Equations ◽  
The Third ◽  
Chebyshev Wavelet ◽  
Fractional Orders

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

A numerical approach for solving nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

2016 ◽  
Vol 14 (05) ◽  
pp. 1650036 ◽  
Author(s):  
P. K. Sahu ◽  
S. Saha Ray
Keyword(s):  
Differential Equations ◽  
Present Method ◽  
Algebraic System ◽  
Numerical Approximation ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Nonlinear Algebraic Equations

In this paper, a numerical approximation based on Legendre wavelets has been developed to solve nonlinear fractional Volterra–Fredholm integro-differential equations. Legendre wavelets are generated by dilation and translation of Legendre polynomials. The properties of the Legendre wavelets are presented in the paper. The proposed wavelet method transforms the integral equations to a system of nonlinear algebraic equations and this algebraic system has been solved numerically by Newton’s method. Convergence analysis of the proposed method has been discussed in this paper. Some examples have been illustrated to show the applicability and accuracy of the present method.

Sinc-Galerkin Technique for the Numerical Solution of Fractional Volterra–Fredholm Integro-Differential Equations with Weakly Singular Kernels

2016 ◽  
Vol 17 (6) ◽  
pp. 315-323 ◽  
Author(s):  
P. K. Sahu ◽  
S. Saha Ray
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Galerkin Method ◽  
Function Approximation ◽  
Present Method ◽  
Sinc Function ◽  
Algebraic Equations ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels

AbstractThe sinc-Galerkin method is developed to approximate the solution of fractional Volterra–Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the sinc function approximation. Usually, this type of integral equations is very difficult to solve analytically as well as numerically. The present method applied to the integral equation reduces to solve the system of algebraic equations. Also the numerical results obtained by sinc-Galerkin method have been compared with the results obtained by existing methods. Illustrative examples have been discussed to demonstrate the validity and applicability of the presented method.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

2021 ◽  
Vol 6 (1) ◽  
pp. 10
Author(s):  
İbrahim Avcı 
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Absolute Error ◽  
Algebraic Equations ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures.

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