Haar wavelet method for solving Fisher’s equation

In this paper, we develop an accurate and efficient Haar wavelet solution of Fisher’s equation, a prototypical reaction-diffusion equation. The solutions of Fisher’s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt Haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.     

A new approach for solving a class of nonlinear integro-differential equations

In this paper, a new technique for solving a class of quadratic integral and integro-differential equations is introduced. The main advantage of this technique is that it can replace the nonlinear problem by an equivalent linear one or by another simpler nonlinear one. The convergence of the series solution is proved. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. Some numerical examples are introduced to verify the efficiency of the new technique.     

A new wavelet method for solving the Helmholtz equation with complex solution

The Helmholtz equation which is very important in a variety of applications, such as acoustic cavity and radiation wave, has been greatly considered in recent years. In this article, we propose a new efficient computational method based on the Legendre wavelets (LWs) expansion together with their operational matrices of integration and differentiation to solve this equation with complex solution. Because of the fact that both of the operational matrices of integration and differentiation are used in the proposed method, the boundary conditions are taken into account automatically. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifies the problems. As an applied example, “propagation of plane waves” is investigated to demonstrate the validity and applicability of the presented method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 741–756, 2016     

A Haar wavelet collocation approach for solving one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations

We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one- and two-dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one- and two-dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate.     

A method for solving the linear boundary value problem for an integro-differential equation

A method for solving the linear boundary value problem for an integro-differential equation is proposed that is based on interval partition and the introduction of additional parameters. Necessary and sufficient conditions for the solvability of the problem are obtained.     

Weak formulation based Haar wavelet method for solving differential equations

The Haar wavelet based discretization method for solving differential equations is developed. Nonlinear Burgers equation is considered as a test problem. Both, strong and weak formulations based approaches are discussed. The discretization scheme proposed is based on the weak formulation. An attempt is made to combine the advantages of the FEM and Haar wavelets. The obtained numerical results have been validated against a closed form analytical solution as well as FEM results. Good agreement with the exact solution has been observed.     

A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order

In this paper we present a computational method for solving a class of nonlinear Fredholm integro-differential equations of fractional order which is based on CAS (Cosine And Sine) wavelets. The CAS wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the technique.     

A wavelet method for solving singular integral equation of MHD

This paper presents Haar wavelet approximation to solve a singular integral equation which has singularities on a diagonal of the domain R=[-1,1]×[-1,1]. The singularities arise basically due to modified Bessel function K₀ which appears as a part of the kernel. Thus the integral equation is weakly (logarithmically) singular only. The problem is solved considering all the singularities of the kernel and results are examined for approximations of different orders. Our interest to solve the problem using Haar wavelet basis is due to its simplicity and efficiency in numerical approximation. The results show convergence trend as mesh is refined. Comparison is made with some available results obtained earlier by partial consideration of the singularities.     

A new approach to solving the stationary Fokker-Planck-Kolmogorov equation for a randomly oscillating nonlinear system

It is shown that the Fokker-Planck-Kolmogorov equation in terms of amplitude and phase may, in the stationary case, be reduced to a first order partial differential equation which we call the stationary reduced Fokker-Planck-Kolmogorov. A method for approximate solution of the reduced equation is presented which does not need assumptions on the smallness of nonlinearity of a system and intensity of random influences.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1123–1129, August, 1992.     

A new method for solving nonlinear differential-difference equation

A new approach is presented which enables one to construct the exact solutions of nonlinear differential difference equations. As its application, the soliton solutions and periodic solutions of Hybrid lattice, discretized mKdV lattice and modified Volterra lattice are conveniently obtained by computing the solutions for a lattice equation introduced by Wadati.     

A new approach for solving the Stokes problem

In this paper, a new approach for finding the approximate solution of the Stokes problem is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a distributed parameter control system, the theory of measure is used to approximate the velocity functions by piecewise linear functions. Then, the approximate values of pressure are obtained by a finite difference scheme.     

A numerical approach for solving an extended Fisher-Kolomogrov-Petrovskii-Piskunov equation

In the present paper a numerical method, based on finite differences and spline collocation, is presented for the numerical solution of a generalized Fisher integro-differential equation. A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed-form difference scheme. A number of test examples are solved to assess the accuracy of the method. The numerical solutions obtained, indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods. Convergence and stability of the scheme have also been discussed.     

A semigroup approach to an integro-differential equation modeling slow erosion

The paper is concerned with a scalar conservation law with nonlocal flux, providing a model for granular flow with slow erosion and deposition. While the solution u=u(t,x)$u=u(t,x)$ can have jumps, the inverse function x=x(t,u)$x=x(t,u)$ is always Lipschitz continuous; its derivative has bounded variation and satisfies a balance law with measure-valued sources. Using a backward Euler approximation scheme combined with a nonlinear projection operator, we construct a continuous semigroup whose trajectories are the unique entropy weak solutions to this balance law. Going back to the original variables, this yields the global well-posedness of the Cauchy problem for the granular flow model.     

A collocation method for solving singular integro-differential equations

This paper describes a collocation method for numerically solving Cauchy-type linear singular integro-differential equations. The numerical method is based on the transformation of the integro-differential equation into an integral equation, and then applying a collocation method to solve the latter. The collocation points are chosen as the Chebyshev nodes. Uniform convergence of the resulting method is then discussed. Numerical examples are presented and solved by the numerical techniques.     

A new direct method for solving the Black–Scholes equation

Using the Mellin transform a new method for solving the Black–Scholes equation is proposed. Our approach does not require either variable transformations or solving diffusion equations.     

A new approach for solving repair limit problems

The usual approach to finding optimal repair limits on failure of a component is to use a finite state approximation Markov Decision Process (MDP). In this paper an alternative approach is introduced. Assuming a stochastically increasing repair cost, the optimum solution is shown to satisfy a certain two-point boundary condition, first order differential equation. An asymptotic formula for the optimal repair limit function is derived. Numerical solutions are obtained for some Weibull and Special Erlang distributed time to failure distributions. The structural form of the repair limit function results in a solution procedure which is several orders of magnitude faster than is achievable using previous methods.     

Optimal quadrature for Haar wavelet spaces

This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, . The asymptotic orders of the errors are derived for the case of the scrambled -nets and -sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands .