A new approach for solving one-dimensional fractional boundary value problems via Haar wavelet collocation method

2021 ◽  
Vol 160 ◽  
pp. 313-330
Author(s):  
KumSong Jong ◽  
HuiChol Choi ◽  
KyongJun Jang ◽  
SunAe Pak
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Haar Wavelet ◽  
New Approach ◽  
One Dimensional ◽  
Wavelet Collocation Method ◽  
Fractional Boundary Value Problems

scholarly journals A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven

PLoS ONE ◽  
2021 ◽  
Vol 16 (1) ◽  
pp. e0244027
Author(s):  
Sidra Saleem ◽  
Malik Zawwar Hussain ◽  
Imran Aziz
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Collocation Method ◽  
Approximate Solutions ◽  
Haar Wavelet ◽  
Partial Differential ◽  
Lax Equation ◽  
One Dimensional ◽  
Wavelet Collocation Method

The approximate solution of KdV-type partial differential equations of order seven is presented. The algorithm based on one-dimensional Haar wavelet collocation method is adapted for this purpose. One-dimensional Haar wavelet collocation method is verified on Lax equation, Sawada-Kotera-Ito equation and Kaup-Kuperschmidt equation of order seven. The approximated results are displayed by means of tables (consisting point wise errors and maximum absolute errors) to measure the accuracy and proficiency of the scheme in a few number of grid points. Moreover, the approximate solutions and exact solutions are compared graphically, that represent a close match between the two solutions and confirm the adequate behavior of the proposed method.

scholarly journals Further results on existence of positive solutions of generalized fractional boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hojjat Afshari ◽  
Mohammed S. Abdo ◽  
Jehad Alzabut
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Operators ◽  
New Approach ◽  
Existence Of Positive Solutions ◽  
Contractive Type ◽  
Fractional Boundary Value Problems

Abstract This paper studies two classes of boundary value problems within the generalized Caputo fractional operators. By applying the fixed point result of α-ϕ-Geraghty contractive type mappings, we derive new results on the existence and uniqueness of the proposed problems. Illustrative examples are constructed to demonstrate the advantage of our results. The theorems reported not only provide a new approach but also generalize existing results in the literature.

An Efficient Wavelet-Based Collocation Method for Handling Singularly Perturbed Boundary-Value Problems in Fluid Mechanics

2017 ◽  
Vol 18 (6) ◽  
pp. 485-494
Author(s):  
Firdous A. Shah ◽  
Rustam Abass
Keyword(s):  
Fluid Mechanics ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Singularly Perturbed ◽  
Test Problems ◽  
Algebraic Equations ◽  
Specific Test

AbstractIn this article, we develop an accurate and efficient wavelet-based collocation method for solving both linear and nonlinear singularly perturbed boundary-value problems that arise in fluid mechanics. The properties of the Haar wavelet expansions together with operational matrix of integration are used to convert the underlying problems into systems of algebraic equations which can be efficiently solved by suitable solvers. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

scholarly journals Efficient Numerical Scheme for the Solution of Tenth Order Boundary Value Problems by the Haar Wavelet Method

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1874
Author(s):  
Rohul Amin ◽  
Kamal Shah ◽  
Imran Khan ◽  
Muhammad Asif ◽  
Mehdi Salimi ◽  
...  
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Haar Wavelet ◽  
Nonlinear Boundary Value Problems ◽  
Haar Functions ◽  
Collocation Points ◽  
Wavelet Collocation Method ◽  
Mean Square Errors ◽  
Gauss Points

In this paper, an accurate and fast algorithm is developed for the solution of tenth order boundary value problems. The Haar wavelet collocation method is applied to both linear and nonlinear boundary value problems. In this technqiue, the tenth order derivative in boundary value problem is approximated using Haar functions and the process of integration is used to obtain the expression of lower order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking validation and the convergence of the proposed technique. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The experimental rate of convergence using different number of collocation points is also calculated, which is nearly equal to 2.

Sign in / Sign up

Export Citation Format

Share Document