A numerical approach for solving weakly singular partial integro-differential equations via two-dimensional-orthonormal Bernstein polynomials with the convergence analysis

2018 ◽  
Vol 35 (2) ◽  
pp. 615-637 ◽  
Author(s):  
Farshid Mirzaee ◽  
Sahar Alipour ◽  
Nasrin Samadyar
Keyword(s):  
Differential Equations ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Numerical Approach ◽  
Two Dimensional ◽  
Weakly Singular

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.

A numerical study of two-dimensional coupled systems and higher order partial differential equations

2019 ◽  
Vol 12 (05) ◽  
pp. 1950071
Author(s):  
R. Rohila ◽  
R. C. Mittal
Keyword(s):  
Differential Equations ◽  
Numerical Study ◽  
Differential Quadrature Method ◽  
Bernstein Polynomials ◽  
Differential Quadrature ◽  
Coupled Systems ◽  
Two Dimensional ◽  
Quadrature Method ◽  
Grid Points ◽  
The Stability

In this paper, a new approach and methodology is developed by incorporating differential quadrature technique with Bernstein polynomials. In differential quadrature method, approximations are done in a way that the derivatives of the function are replaced by a linear sum of functional values at the grid points of the given domain. In Bernstein differential quadrature method (BDQM), Bernstein polynomials are employed for spatial discretization so that a system of ordinary differential equations (ODE’s) is obtained which is solved by SSPRK-43 method. The stability of the method is also studied. The accuracy of the present method is checked by performing numerical experiments on two-dimensional coupled Burgers’ and Brusselator systems and fourth-order extended Fisher Kolmogorov (EFK) equation. Implementation of the method is very easy, efficient and capable of reducing the size of computational efforts.

scholarly journals Solutions of Volterra integral and integro-differential equations using modified Laplace Adomian decomposition method

2019 ◽  
Vol 15 (1) ◽  
pp. 5-18
Author(s):  
D. Rani ◽  
V. Mishra
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Convergence Analysis ◽  
Decomposition Method ◽  
Bernstein Polynomials ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Modified Technique ◽  
Adomian Decomposition

Abstract In this paper, an effectual and new modification in Laplace Adomian decomposition method based on Bernstein polynomials is proposed to find the solution of nonlinear Volterra integral and integro-differential equations. The performance and capability of the proposed idea is endorsed by comparing the exact and approximate solutions for three different examples on Volterra integral, integro-differential equations of the first and second kinds. The results shown through tables and figures demonstrate the accuracy of our method. It is concluded here that the non orthogonal polynomials can also be used for Laplace Adomian decomposition method. In addition, convergence analysis of the modified technique is also presented.

Sign in / Sign up

Export Citation Format

Share Document