Novel Functional Matrix Method using Standard Basis of Polynomial Linear Space

This paper has developed a novel functional matrix using the standard basis of (n+1) dimensional polynomial linear space to solve second-order singular initial and boundary problems. The linearly independent polynomials properties are used to convert the differential equations into algebraic equations with suitable solvers that can efficiently solve. Seven numerical examples are considered to demonstrate this technique's applicability and efficiency. The obtained results are compared favorably with the exact solutions. Also, we proved some theorems on convergence, exact solutions, and uniform convergence

A New Approach for the Numerical Solution for Nonlinear Klein–Gordon Equation

In this article, we generated a new operational matrix of integration using Clique polynomials of complete graphs and also introducing a new numerical technique to solve nonlinear Klein–Gordon equation. These equations describe a variety of physical phenomena such as ferroelectric and ferromagnetic domain walls, and DNA dynamics. We obtain an approximate solution for the nonlinear Klein–Gordon equation using the present method by transforming a system of nonlinear algebraic equations. The proposed scheme is applied to some examples and compared with another method in the literature that demonstrates the effectiveness of this method

Numerical solution for the (2+1) dimensional Sobolev and regularized long wave equations arise in fluid mechanics via wavelet technique

This article proposed an efficient numerical technique for the solution of (2+1) dimensional Sobolev and regularized long wave equations that arise in fluid mechanics using the Laguerre wavelet collocation method. Five examples are illustrated to inspect the proposed technique efficiency, and convergence analysis is discussed in terms of a theorem. Here, the Sobolev and regularized long wave equations are converted into a system of algebraic equations using the properties of Laguerre wavelet, and solutions obtained by the proposed scheme are more accurate, and they are compared with the analytical solution and other methods in the literature by calculating L2 and L∞ Errors. © 2020 The Author(s

Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems

In this paper, new operational matrix of integration is generated by using Hermite wavelets. By aid of these matrices, Hermite wavelets operational matrix method (HWOMM) is developed for second ordered nonlinear singular initial value problems. Properties of Hermite wavelets are used to convert the differential equations into system of algebraic equations which can be efficiently solved by suitable solvers. Illustrative numerical problems are considered to demonstrate the applicability of the proposed technique and those results are comparing favourably with the exact solutions and errors. Keywords: Hermite wavelets, Nonlinear singular initial value problems, Collocation method, Operational matri

Hermite wavelet method for solving nonlinear Rosenau–Hyman equation

In this paper, we present an approximate solution for solving the nonlinear Rosenau–Hyman equation. The method is based on adapting the wavelet technique accompanied with the Hermit polynomials. Convergence analysis for the proposed method is being investigated, proving that the Hermite wavelet expansion is uniformly convergent. A test example is presented for different values of the parameters, and the obtained results are compared to other relevant methods from the literature. The process proves to have the ability to produce accurate results than the other compared methods. Some graphical representations for the problem are drawn to illustrate the behavior of the solution

A Novel Approach on Micropolar Fluid Flow in a Porous Channel with High Mass Transfer via Wavelet Frames

In this article, we present the Laguerre wavelet exact Parseval frame method (LWPM) for the two-dimensional flow of a rotating micropolar fluid in a porous channel with huge mass transfer. This flow is governed by highly nonlinear coupled partial differential equations (PDEs) are reduced to the nonlinear coupled ordinary differential equations (ODEs) using Berman's similarity transformation before being solved numerically by a Laguerre wavelet exact Parseval frame method. We also compared this work with the other methods in the literature available. Moreover, in the graphs of the velocity distribution and microrotation, we shown that the proposed scheme's solutions are more accurate and applicable than other existing methods in the literature. Numerical results explaining the effects of various physical parameters connected with the flow are discussed

Numerical Solution of Abel′s Integral Equations using Hermite Wavelet

A numerical method is developed for solving the Abel′s integral equations is presented. The method is based upon Hermite wavelet approximations. Hermite wavelet method is then utilized to reduce the Abel′s integral equations into the solution of algebraic equations. Illustrative examples are included to demonstrate the validity, efficiency and applicability of the proposed technique. Algorithm provides high accuracy and compared with other existing methods

Modified Bernoulli wavelets functional matrix approach for the HIV infection of CD4+ T cells model

In this study, we generated a novel functional matrix using Bernoulli wavelets. Also, we developed a novel technique called the Bernoulli wavelets collocation method to obtain reasonably accurate solutions for the HIV-infection model of CD4+ T cells. This mathematical model is in the form of a system of a nonlinear ordinary differential equation (ODE). This approach obtains the solution for this model by transforming it into a system of nonlinear algebraic equations by expanding through Bernoulli wavelets with unknown coefficients. The collocation scheme is used to calculate these unknown coefficients. The consistency and proficiency of the developed approach are demonstrated through tables and graphs. Obtained results reveal that the current approach is more accurate than other methods in the literature. All computations have been made with the help of Mathematica software. Some properties of Bernoulli wavelets are discussed in terms of theorems

The new operational matrix of integration for the numerical solution of integro-differential equations via Hermite wavelet

Hermite wavelet method (HWM) is applied to approximate the solution of the integro-differential equations. The Hermite wavelet and operational matrix of integration are used to reduce the integro-differential equations into algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples. © 2021, The Author(s), under exclusive licence to Sociedad Española de Matemática Aplicada

Bernoulli wavelets functional matrix technique for a system of nonlinear singular Lane Emden equations

In the present paper, we developed the functional matrix of integration via Bernoulli wavelets and generated a competent numerical scheme to solve the nonlinear system of singular differential equations which is Lane Emden form by Bernoulli wavelets collocation technique (BWCT) with different physical conditions. The system of nonlinear singular models is not smooth to operate as they are singular and nonlinear. This approach obtains the solution for this system by transforming it into an a-nonlinear system of algebraic equations by expanding through Bernoulli wavelets with unknown coefficients. These unknown coefficients are calculated using the collocation scheme. The consistency and proficiency of the developed approach are demonstrated via graphs and tables. Attained results confirm that the newly implemented approach is more effective and accurate than other techniques which are available in the literature. All computations have been made using Mathematica software. The convergence of this method is explained in terms of theorems