scholarly journals On the Solutions of Fractional Burgers-Fisher and Generalized Fisher’s Equations Using Two Reliable Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
A. K. Gupta ◽  
S. Saha Ray
Keyword(s):  
Exact Solutions ◽  
Asymptotic Method ◽  
Arbitrary Order ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Haar Wavelet ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Optimal Homotopy Asymptotic Method ◽  
Reliable Methods

Two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), are presented. Haar wavelet method is an efficient numerical method for the numerical solution of arbitrary order partial differential equations like Burgers-Fisher and generalized Fisher equations. The approximate solutions thus obtained for the fractional Burgers-Fisher and generalized Fisher equations are compared with the optimal homotopy asymptotic method as well as with the exact solutions. Comparison between the obtained solutions with the exact solutions exhibits that both the featured methods are effective and efficient in solving nonlinear problems. The obtained results justify the applicability of the proposed methods for fractional order Burgers-Fisher and generalized Fisher’s equations.

The comparison of two reliable methods for the accurate solution of fractional Fisher type equation

2017 ◽  
Vol 34 (8) ◽  
pp. 2598-2613
Author(s):  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Type Equation ◽  
Nonlinear Problems ◽  
Haar Wavelet ◽  
Partial Differential ◽  
Wavelet Method ◽  
Content Type ◽  
Haar Wavelet Method ◽  
Optimal Homotopy Asymptotic Method

Purpose The purpose of this paper is the comparative analysis of Haar Wavelet Method and Optimal Homotopy Asymptotic Method for fractional Fisher type equation. In this paper, two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), have been presented. The Haar wavelet method is an efficient numerical method for the numerical solution of fractional order partial differential equation like the Fisher type. The approximate solutions of the fractional Fisher-type equation are compared with those of OHAM and with the exact solutions. Comparisons between the obtained solutions with the exact solutions exhibit that both the featured methods are effective and efficient in solving nonlinear problems. However, the results indicate that OHAM provides more accurate value than the Haar wavelet method. Design/methodology/approach Comparisons between the solutions obtained by the Haar wavelet method and OHAM with the exact solutions exhibit that both featured methods are effective and efficient in solving nonlinear problems. Findings The comparative results indicate that OHAM provides a more accurate value than the Haar wavelet method. Originality/value In this paper, two reliable techniques, the Haar wavelet method and OHAM, have been proposed for solving nonlinear fractional partial differential equation, i.e. fractional Fisher-type equation. The proposed novel methods are well suited for only nonlinear fractional partial differential equations. It also exhibits that the proposed method is a very efficient and powerful technique in finding the solutions for the nonlinear time fractional differential equations. The main significance of the proposed method is that it requires less amount of computational overhead in comparison to other numerical and analytical approximate methods. The application of the proposed methods for the solutions of time fractional Fisher-type equations satisfactorily justifies its simplicity and efficiency.

scholarly journals Haar wavelet collocation method for linear first order stiff differential equations

2020 ◽  
Vol 34 ◽  
pp. 03001
Author(s):  
Mehmet Tarık Atay ◽  
Onur Metin Mertaslan ◽  
Musa Kasım Ağca ◽  
Abdülkadir Yılmaz ◽  
Batuhan Toker
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Haar Wavelet ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
First Order ◽  
Stiff Ordinary Differential Equations ◽  
Wavelet Collocation Method ◽  
Computational Procedures ◽  
Wavelet Family

In general, there are countless types of problems encountered from different disciplines that can be represented by differential equations. These problems can be solved analytically in simpler cases; however, computational procedures are required for more complicated cases. Right at this point, the wavelet-based methods have been using to compute these kinds of equations in a more effective way. The Haar Wavelet is one of the appropriate methods that belongs to the wavelet family using to solve stiff ordinary differential equations (ODEs). In this study, The Haar Wavelet method is applied to stiff differential problems in order to demonstrate the accuracy and efficacy of this method by comparing the exact solutions. In comparison, similar to the exact solutions, the Haar wavelet method gives adequate results to stiff differential problems.

scholarly journals APPLICATION OF HIGHER ORDER HAAR WAVELET METHOD FOR SOLVING NONLINEAR EVOLUTION EQUATIONS

2020 ◽  
Vol 25 (2) ◽  
pp. 271-288 ◽  
Author(s):  
Mart Ratas ◽  
Andrus Salupere
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Haar Wavelet ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Wavelet Method ◽  
Haar Wavelet Method

The recently introduced higher order Haar wavelet method is treated for solving evolution equations. The wave equation, the Burgers’ equations and the Korteweg-de Vries equation are considered as model problems. The detailed analysis of the accuracy of the Haar wavelet method and the higher order Haar wavelet method is performed. The obtained results are validated against the exact solutions.

scholarly journals Solving Nonlinear Boundary Value Problems Using the Higher Order Haar Wavelet Method

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2809
Author(s):  
Mart Ratas ◽  
Jüri Majak ◽  
Andrus Salupere
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Boundary ◽  
Haar Wavelet ◽  
Higher Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Nonlinear Boundary Value Problems ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Liénard Equations

The current study is focused on development and adaption of the higher order Haar wavelet method for solving nonlinear ordinary differential equations. The proposed approach is implemented on two sample problems—the Riccati and the Liénard equations. The convergence and accuracy of the proposed higher order Haar wavelet method are compared with the widely used Haar wavelet method. The comparison of numerical results with exact solutions is performed. The complexity issues of the higher order Haar wavelet method are discussed.

Approximate Solutions to a Cantilever Beam Using Optimal Homotopy Asymptotic Method

2013 ◽  
Vol 430 ◽  
pp. 22-26 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herisanu ◽  
Traian Marinca
Keyword(s):  
Small Parameter ◽  
Cantilever Beam ◽  
Asymptotic Method ◽  
Approximate Solutions ◽  
Lumped Mass ◽  
Engineering Problems ◽  
Approximate Frequency ◽  
Optimal Homotopy Asymptotic Method ◽  
Analytical Expressions

The response of a cantilever beam with a lumped mass attached to its free end subject to harmonical excitation at the base is investigated by means of the Optimal Homotopy Asymptotic Method (OHAM). Approximate accurate analytical expressions for the solutions and for approximate frequency are determined. This method does not require any small parameter in the equation. The obtained results prove that our method is very accurate, effective and simple for investigation of such engineering problems.

scholarly journals A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations

2018 ◽  
Vol 3 (2) ◽  
pp. 447-458 ◽  
Author(s):  
S.C. Shiralashetti ◽  
H. S. Ramane ◽  
R.A. Mundewadi ◽  
R.B. Jummannaver
Keyword(s):  
Error Analysis ◽  
Comparative Study ◽  
Numerical Solution ◽  
Integral Equations ◽  
Haar Wavelet ◽  
Fredholm Integral Equations ◽  
Polynomial Method ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Hosoya Polynomial

AbstractIn this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.

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