Wavelet Method for a Class of Fractional Klein-Gordon Equations

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
G. Hariharan
Keyword(s):  
Applied Mathematics ◽  
Operational Matrix ◽  
Correlation Coefficients ◽  
Haar Wavelet ◽  
Mathematical Tool ◽  
Algebraic Equations ◽  
Fundamental Idea ◽  
Wavelet Method ◽  
Klein Gordon ◽  
Constant Coefficients

Wavelet analysis is a recently developed mathematical tool in applied mathematics. A wavelet method for a class of space-time fractional Klein–Gordon equations with constant coefficients is proposed, by combining the Haar wavelet and operational matrix together and efficaciously dispersing the coefficients. The behavior of the solutions and the effects of different values of fractional order α are graphically shown. The fundamental idea of the Haar wavelet method is to convert the fractional Klein–Gordon equations into a group of algebraic equations, which involves a finite number of variables. The examples are given to demonstrate that the method is effective, fast, and flexible; in the meantime, it is found that the difficulties of using the Daubechies wavelets for solving the differential equation, which need to calculate the correlation coefficients, are avoided.

Haar wavelet method for the numerical solution of Klein–Gordan equations

2016 ◽  
Vol 09 (01) ◽  
pp. 1650012 ◽  
Author(s):  
S. C. Shiralashetti ◽  
L. M. Angadi ◽  
A. B. Deshi ◽  
M. H. Kantli
Keyword(s):  
Good Accuracy ◽  
Haar Wavelet ◽  
Quantum Field ◽  
Algebraic Equations ◽  
Applied Physics ◽  
Fundamental Idea ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Klein Gordon ◽  
Almost All

Wavelets have become a powerful tool for having applications in almost all the areas of engineering and science such as numerical simulation of partial differential equations. In this paper, we present the Haar wavelet method (HWM) to solve the linear and nonlinear Klein–Gordon equations which occur in several applied physics fields such as, quantum field theory, fluid dynamics, etc. The fundamental idea of HWM is to convert the Klein–Gordon equations into a group of algebraic equations, which involve a finite number of variables. The examples are given to demonstrate the numerical results obtained by HWM, are compared with already existing numerical method i.e. finite difference method (FDM) and exact solution to confirm the good accuracy of the presented scheme.

scholarly journals Haar wavelets approach of traveling wave equation- A plausible solution of lightning stroke model

2013 ◽  
Vol 2 (2) ◽  
pp. 149
Author(s):  
Hariharan Gopalakrishnan ◽  
R. Rajaraman Raman ◽  
K. Kannan Kirthivasan
Keyword(s):  
Traveling Wave ◽  
Wave Model ◽  
Haar Wavelet ◽  
Haar Wavelets ◽  
Algebraic Equations ◽  
Lightning Stroke ◽  
Fundamental Idea ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Manageable Method

This paper describes a traveling wave model for describing the lightning stroke by the Haar wavelet method (HWM) is proposed. Numerical example is included and illustrated for applicability and validity of the proposed method. The fundamental idea of Haar wavelet method is to convert the differential equations into a group of algebraic equations that involves a finite number of variables. The power of the manageable method is confirmed. The results show that the proposed way is quite reasonable when compared to exact solution. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.

scholarly journals An operational Haar wavelet method for solving fractional Volterra integral equations

2011 ◽  
Vol 21 (3) ◽  
pp. 535-547 ◽  
Author(s):  
Habibollah Saeedi ◽  
Nasibeh Mollahasani ◽  
Mahmoud Moghadam ◽  
Gennady Chuev
Keyword(s):  
Integral Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Global Error Bound ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Abel Integral Equations

An operational Haar wavelet method for solving fractional Volterra integral equationsA Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the validity and applicability of the developed method.

Haar wavelet operational matrix method for system of fractional nonlinear differential equations

2017 ◽  
Vol 15 (05) ◽  
pp. 1750043 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Reliable Method ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Implementation Process ◽  
Haar Wavelet ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

scholarly journals Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
F. M. Alharbi ◽  
A. M. Zidan ◽  
Muhammad Naeem ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Differential Equations ◽  
Fractional Integral ◽  
Numerical Technique ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Haar Wavelet ◽  
Mathematical Tool ◽  
Suggested Technique ◽  
Operational Matrix Of Integration

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

scholarly journals Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Abdulnasir Isah ◽  
Chang Phang ◽  
Piau Phang
Keyword(s):  
Fractional Derivative ◽  
Collocation Method ◽  
Upper Bound ◽  
Present Method ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Mathematical Tool ◽  
Algebraic Equations ◽  
Genocchi Polynomials

An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.

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.

New wavelet method for solving boundary value problems arising from an adiabatic tubular chemical reactor theory

2020 ◽  
Vol 13 (07) ◽  
pp. 2050059
Author(s):  
Mohamed R. Ali ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Solutions ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Chemical Reactor ◽  
Steady State Solution ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Newton’S Iterative Method ◽  
Present Technique ◽  
Exothermic Chemical Reaction

This paper displays an efficient numerical technique of realizing mathematical models for an adiabatic tubular chemical reactor which forms an irreversible exothermic chemical reaction. At a steady-state solution for an adiabatic rounded reactor, the model can be diminished to a conventional nonlinear differential equation which converts into a system of the nonlinear equation that can proceed numerically utilizing Newton’s iterative method. An operational matrix of coordination is derived and is utilized to decrease the model for an adiabatic tubular chemical reactor to an arrangement of algebraic equations. Simple execution, basic activities, and precise arrangements are the fundamental highlights of the proposed wavelet technique. The numerical solutions attained by the present technique have been contrasted and compared with other techniques.

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.

scholarly journals Legendre wavelet operational matrix method for solving fractional differential equations in some special conditions

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 203-214
Author(s):  
Aydin Secer ◽  
Selvi Altun ◽  
Mustafa Bayram
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Mathematical Tool ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Operational Matrix Method ◽  
Fractional Differential ◽  
A New Technique

This paper proposes a new technique which rests upon Legendre wavelets for solving linear and non-linear forms of fractional order initial and boundary value problems. In some particular circumstances, a new operational matrix of fractional derivative is generated by utilizing some significant properties of wavelets and orthogonal polynomials. We approached the solution in a finite series with respect to Legendre wavelets and then by using these operational matrices, we reduced the fractional differential equations into a system of algebraic equations. Finally, the introduced technique is tested on several illustrative examples. The obtained results demonstrate that this technique is a very impressive and applicable mathematical tool for solving fractional differential equations.

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