Two Dimensional Wavelets Collocation Scheme for Linear and Nonlinear Volterra Weakly Singular Partial Integro-Differential Equations

Patel, Vijay Kumar; Singh, Suryabhan; Singh, Vineet Kumar; Tohidi, Emran

doi:10.1007/s40819-018-0560-4

Cited by 20 publications

(19 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem The series

\sum_{n = 1}^{2^{k - 1}} \sum_{m = 0}^{\infty} \sum_{n^{'} = 1}^{2^{k^{'} - 1}} \sum_{m^{'} = 0}^{\infty} c_{nm n^{'} m^{'}} Φ false(a, b false)

approximation of f ( a , b ) using 2D Legendre wavelet or 2D Bernoulli wavelet converges to f ( a , b ). Proof See [17]. Remark If f N ( a , b ) be N th approximation of f ( a , b ) then by above theorem, we can conclude that

‖ f false(a, b false) - f_{N} false(a, b false) ‖ \to 0, as N \to \infty .

…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Numerical wavelets scheme to complex partial differential equation arising from Morlet continuous wavelet transform

Patel

Singh

2020

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

In this article, a new wavelets approximation scheme based on operational matrices has been discussed for the solutions of proposed initial value complex partial differential equation (CPDE) which is arising from continuous wavelet transform. We developed an algorithm using transformation to obtain the numerical solution of CPDE with a numerical wavelet collocation method based on two‐dimensional Legendre wavelet (TDLW) and two‐dimensional Bernoulli wavelet (TDBW) operational matrices. We converted CPDE into a system of partial differential equations (PDEs) and this system of PDEs converted into couple PDEs in terms of the real and imaginary component. Finally, we obtained a system of partial integro‐differential equations (PIDEs) with nonlocal boundary conditions. After implementation of wavelets scheme on PIDEs, PIDEs reduced into a system of algebraic equations. In the sequence of this, we also investigated the convergence analysis of TDLW, TDBW, and error analysis, respectively. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented wavelet scheme for the proposed problem.

show abstract

“…Theorem The series

\sum_{n = 1}^{2^{k - 1}} \sum_{m = 0}^{\infty} \sum_{n^{'} = 1}^{2^{k^{'} - 1}} \sum_{m^{'} = 0}^{\infty} c_{nm n^{'} m^{'}} Φ false(a, b false)

‖ f false(a, b false) - f_{N} false(a, b false) ‖ \to 0, as N \to \infty .

…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Numerical wavelets scheme to complex partial differential equation arising from Morlet continuous wavelet transform

Patel

Singh

2020

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

show abstract

“…Wavelet has been used extensively during the last few years in solving different types of differential equations. Many mathematicians' contributions to wavelet‐based numerical methods are as follows, for example, in solving Benjamin–Bona–Mahony equations, 15 nonlinear singular initial value problems, 16 generalized Burgers–Huxley equation, 17 nonlinear Lane–Emden‐type equations, 18 and nonlinear Volterra weakly singular partial integrodifferential equations 19 . For further details about the use of wavelet‐based methods using different bases, see several studies 20–24 .…”

Section: Introductionmentioning

confidence: 99%

An effective numerical simulation for solving a class of Fokker–Planck equations using Laguerre wavelet method

Kumbinarasaiah

Rezazadeh

Adel

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

This work presents a fast and efficient approach for solving the Fokker–Planck equation via the Laguerre wavelet collocation method. Properties of the Laguerre wavelet are presented, along with the Laguerre basis to reduce the problems into systems of either linear or nonlinear algebraic equations. This system is then solved to find the unknown coefficients. A convergent analysis for the proposed scheme is explained through theorems. Several numerical examples are presented to demonstrate the applicability and accuracy of the method to a wide variety of similar problems. The results are compared to other methods through graphs and tables.

show abstract

“…T is the terminal time, and Ω denotes a bounded domain with boundary Γ, where Ω ⊂ ℝ d with d = 1, 2, 3. This kind partial integro-differential equation is indicated by Volterra [1]; the new results of Volterra partial integro-differential equation can be found in [2,3]. The problem Equation (1) can be found in the model of heat flow in a conducting materials with memory [4][5][6][7] and in viscoelastic mechanics [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Two-Level Mesh Method for Partial Integro-Differential Equation

Tang

Luo

Zhang

et al. 2022

Journal of Function Spaces

View full text Add to dashboard Cite

In this paper, we present two-level mesh scheme to solve partial integro-differential equation. The proposed method is based on a finite difference method. For the first step, we use finite difference method in time and global radial basis function (RBF) finite difference (FD) in space. For the second step, we use the finite difference method to solve the proposed problem. This two-level mesh scheme is obtained by combining the radial basis function with finite difference. We prove the stability and convergence of scheme and show that the convergence order is O τ 2 + h 2 , where τ and h are the time step size and space step size, respectively. The results of numerical examples are compared with analytical solutions to show the efficiency of proposed scheme. The numerical results are in good agreement with theoretical ones.

show abstract

Two Dimensional Wavelets Collocation Scheme for Linear and Nonlinear Volterra Weakly Singular Partial Integro-Differential Equations

Cited by 20 publications

References 39 publications

Numerical wavelets scheme to complex partial differential equation arising from Morlet continuous wavelet transform

Numerical wavelets scheme to complex partial differential equation arising from Morlet continuous wavelet transform

An effective numerical simulation for solving a class of Fokker–Planck equations using Laguerre wavelet method

Analysis of Two-Level Mesh Method for Partial Integro-Differential Equation

Contact Info

Product

Resources

About