Operational Matrices Of Integration Research Articles

A numerical scheme for fractional order mortgage model of economics

In this article, the famous mortgage model of economics is investigated by developing a numerical scheme. The considered model is proposed under the Caputo power law derivative of fractional order. Further, in the considered model by utilizing the time esteem of cash rule, we build up break even with vital and interest mortgage model that gives quickest installment plan with the least interest rate. The proposed scheme is based on some operational matrices of integration and differentiation of fractional order. For the required operational matrices, we use shifted Legendre polynomials. With the help of the operational matrices, we establish a numerical algorithm to convert the considered model to a system of Lyapunov matrix equation. By using Matlab, we then solve the resultant algebraic equation to get the required solution in numerical form. Further, we plot the approximate solution for various fractional order graphically.

NUMERICAL ASSESSMENT OF THE BRAIN TUMOR GROWTH MODEL VIA FIBONACCI AND HAAR WAVELETS

The main goal of this paper is to present a novel numerical scheme based on the Fibonacci wavelets for solving the brain tumor growth model governed by the Burgess equation. At the first instance, the Fibonacci-wavelet-based operational matrices of integration are obtained by following the well-known Chen–Hsiao technique. These matrices play a vital role in converting the said model into an algebraic system, which could be handled with any standard numerical method. To access the effect of medical treatment over the brain tumor growth, we have investigated both the linear and nonlinear cases of Burgess equation. The nonlinearity arising in the Burgess equation is handled by invoking the quasilinearization technique. In order to compare the efficiency of the Fibonacci-wavelet-based numerical technique, we formulated an analogous numerical scheme based on the Haar wavelets. Subsequently, both the methods are testified on several test problems and it is demonstrated that the Fibonacci wavelet method yields a much more stable solution and a better approximation than the Haar wavelet method.

Operational matrix approach based on two-dimensional Boubaker polynomials for solving nonlinear two-dimensional integral equations

Two-dimensional First Boubaker polynomials (2D-FBPs) have been formulated and developed as the set of basis for the expansion of bivariate functions. These polynomials are an extension of one-dimensional (1D) FBPs that have previously been used to solve 1D nonlinear integral equations. The dual operational matrix, four operational matrices for integration and operational matrix of product for 2D-FBPs are presented in this study newly. The explicit formula for the operational matrix of product is based on the operational matrix of product by 2D Taylor polynomials (TPs), which has been determined too. Moreover, 2D-FBPs and their operational matrices are used to approximate solution of four types of 2D nonlinear integral equations, that are 2D nonlinear Fredholm, Volterra, Volterra–Fredholm and the system of Volterra integral equations. Using Newton–Cotes collocation nodes, our approach leads to the solution of the set of nonlinear algebraic equations and can be solved uniquely by an iteration method. Since the proposed approach does not require any integration, so all the computations can be easily performed. An explicit and simpler upper bound is obtained for the error vector function of the operational matrix of integration P2Db. In addition, an explicit and simpler upper bound is presented for each bivariate function that expand by 2D-FBPs, as well as the convergence analysis has been proved. Six test problems are given to verify the validity and applicability of the presented methods. The numerical results are compared with exciting methods in literature.

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

In this work, we present a spectral method for recovering an unknown time‐dependent lower‐order coefficient and unknown wave displacement in a nonlinear Klein–Gordon equation with overdetermination at a boundary condition. We apply the initial and boundary conditions to construct the satisfier function and use this function in a transformation to convert the main problem to a nonclassical hyperbolic equation with homogeneous initial and boundary conditions. Then, we utilize the orthonormal Bernstein basis functions to approximate the solution of the reformulated problem and use a direct technique based on the operational matrices of integration and differentiation of these basis functions together with the collocation technique to reduce the problem to a system of nonlinear algebraic equations. Regarding the perturbed measurements, the method takes advantage of the mollification method in order to derive stable numerical derivatives. Numerical simulations for solving several test examples are presented to show the applicability of the proposed method for obtaining accurate and stable results.

Composition of Euler Scaling Functions with the Optimization Method for Fractional Hyperbolic and Reaction-Diffusion Equations with Nonlocal Boundary Conditions

This paper provides a new optimization technique for solving fractional hyperbolic and reaction-diffusion equations with nonlocal boundary conditions. Base on Euler scaling functions (ESFs) and operational matrices of integration and derivative, we develop the optimization method to get the approximate solution with high precision. In the process of the numerical algorithm, by applying the operational matrices, we transform the problems into a system of algebraic equations. It should be noted that we investigate and analyze the method error. Eventually, to confirm the applicability of the explained approach, we examine several examples and demonstrate the results in tables and graphical curves.

Piecewise barycentric interpolating functions for the numerical solution of Volterra integro‐differential equations

This investigation presents an effective numerical scheme using a new set of basis functions, namely, the piecewise barycentric interpolating functions, to find the approximate solution of Volterra integro‐differential equations (VIDEs). The operational matrices of integration and product for the PBIFs are provided. Then these operational matrices are utilized to reduce the VIDEs to a system of algebraic equations. Applying the Floater–Hormann weights, the convergence analysis of the PBIFs method is studied. Finally, several numerical examples are provided to illustrate the efficiency and validity of the proposed method in acceptable computational times, and the results are compared with some existing numerical methods.

Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations

In this article, an effective approach has been proposed to obtain the approximate solutions of linear and nonlinear two-dimensional Volterra integro-differential equations. First, the two-dimensional wavelets are introduced, and using it the operational matrices of integration, differentiation, and product have been constructed. Then, by utilizing the properties and matrices of wavelets along with the collocation point, the matrix form of Volterra integro-differential equations has been derived. This approach reduces the two-dimensional linear and nonlinear Volterra integro-differential equations into the system of linear and nonlinear algebraic equations respectively. The convergence analysis and error analysis have been extensively studied by the help of two-dimensional wavelets approximation. Some illustrative examples are examined to clarify the accuracy and effectiveness of the proposed scheme. The graphical representations obtained by two proposed wavelets have been plotted to justify the applicability and validity of the method.

Numerical solution of two-dimensional nonlinear Volterra integral equations of the first kind using operational matrices of hybrid functions

In this paper, operational matrices of two-dimensional hybrid of block-pulse functions and Legendre polynomials are employed to approximate the solutions of two-dimensional nonlinear Volterra integral equations of the first kind (2DNVIEF). To this aim, we first convert 2DNVIEF to the two-dimensional linear Volterra integral equations of the second kind (2DLVIES). Then the operational matrices of integration, together with the product operational matrix, can be applied to reduce the 2DLVIES to a system of algebraic equations. The convergence of the proposed method is studied, and an error estimation under the -norm is provided. Finally, some numerical examples are prepared to clarify the accuracy and efficiency of the numerical method especially in dealing with functions that are not continuous in the whole interval.

Numerical solutions for linear ordinary differential equation with variable coefficients using Haar wavelet method

Recently, wavelet analysis or wavelet transform has been developed as a mathematical instrument for many problems. In the present paper, the numerical technique is based on Haar wavelet collocation points and operational matrices over a new interval [-ϑ,ϑ]. So, we have established a new operational matrix of integration the Haar wavelets functions for a chosen domain from -ϑ to 0. The operational matrices of integration and product are utilized to convert the linear ordinary differential equations together with variable coefficients into algebraic matrix equation which can be resolved by MATLAB. Three numerical examples have been displayed which involving first order differential equations including variable coefficients. The results illustrate that the suggested approach is completely reasonable in comparison with correct solution. Moreover, the exactness of the variable can be enhanced by increasing the Haar wavelets resolution.

Hermite wavelet method for solving oscillatory electrical circuit equations

In this research, numerical solutions based on Hermite wavelets have been presented for solving oscillatory electrical circuit equations. The proposed numerical technique is based on Hermite wavelet basis functions, operational matrices of integration and collocation points. Numerical experiments have been performed to illustrate the accuracy and efficiency of the proposed scheme.

Some properties of orthogonal linear splines and their applications to inverse problems

In this paper, an orthogonal basis for the space of linear splines based on linear B-splines is introduced. Some properties and modifications of this basis are investigated, then operational matrices of integration in collocation points are obtained using stable formula. Theoretical considerations are discussed. Applications to the numerical solutions for some linear and nonlinear inverse problems are given, in which the approximations are obtained using the first and second integrals of orthogonal linear splines that lead to an efficient solution procedure. For solving linear problems, the proposed method is combined with the Tikhonov regularization. Also, the trust-region-dogleg method is used for nonlinear equations. Numerical results validate the effectiveness of the orthogonal linear spline basis for linear and nonlinear inverse problems.

Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

The inverse problem of identifying the diffusion coefficient in the one‐dimensional parabolic heat equation is studied. We assume that the information of Dirichlet boundary conditions along with an integral overdetermination condition is available. By applying the given assumptions, the problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. Then we employ the direct technique based on the operational matrices for integration, differentiation, and the product of the orthonormal polynomials together with the Ritz–Galerkin technique to reduce the main problem to the solution of a system of nonlinear algebraic equations. Numerical simulations are presented to show the applicability of the proposed method.

A novel wavelets operational matrix method for the time variable-order fractional mobile–immobile advection–dispersion model

A novel computational technique for the solution of the variable-order fractional mobile–immobile advection–dispersion equation has been presented in this paper. Firstly, operational integration matrices and variable-order fractional derivatives were deduced using Boubaker wavelets to implement this proposed technique. Utilizing Boubaker wavelets basis for functions approximations and the operational matrices of integration and variable-order fractional derivative along with collocation points, the variable-order fractional mobile–immobile advection–dispersion equation is reduced into the system of algebraic equations. In addition, to determine the convergence analysis and error estimate of the proposed numerical technique, some useful theorems are discussed. Finally, to analyze the computational efficiency and applicability of the proposed numerical method, several numerical examples are presented.

Numerical technique for higher-order differential equation with variable coefficients via Haar wavelet method

Lately, Haar wavelet has been established as a mathematical implement for several problems. In this article, the numerical method is founded on operational matrices and Haar wavelet collocation points on a new-found interval [0, Ω]. For a chosen domain from 0 to Ω, we have established the method of integration the Haar wavelets functions. The operational matrices of integration and product are used to change the higher order differential equations containing variable coefficients into system of algebraic equations that can be solved via MATLAB. Three numerical cases have be presented which including higher order linear differential equations involving variable coefficients. The numerical solution completely converge in comparison with accurate results.

A New Solution for The Enzymatic Glucose Fuel Cell Model with Morrison Equation via Haar Wavelet Collocation Method

Integral equations are essential tools in various areas of applied mathematics. A computational approach to solving an integral equation is important in scientific research. The Haar wavelet collocation method (HWCM) with operational matrices of integration is one famous method which has been applied to solve systems of linear integral equations. In this paper, an approximated analytical method based on the Haar wavelet collocation method is applied to the system of diffusion convection partial differential equations with initial and boundary conditions. This system determines the enzymatic glucose fuel cell with the chemical reaction rate of the Morrison equation. The enzymatic glucose fuel cell model describes the concentration of glucose and hydrogen ion that can be converted into energy. During the process, the model reduces to the linear integral equation system including computational Haar matrices. The computational Haar matrices can be computed by HWCM coding in the Maple program. Illustrated examples are provided to demonstrate the preciseness and effectiveness of the proposed method. The results are shown as numerical solutions of glucose and hydrogen ion.

Bernoulli wavelet method for numerical solution of anomalous infiltration and diffusion modeling by nonlinear fractional differential equations of variable order

In this paper, generalized fractional-order Bernoulli wavelet functions based on the Bernoulli wavelets are constructed to obtain the numerical solution of problems of anomalous infiltration and diffusion modeling by a class of nonlinear fractional differential equations with variable order. The idea is to use Bernoulli wavelet functions and operational matrices of integration. Firstly, the generalized fractional-order Bernoulli wavelets are constructed. Secondly, operational matrices of integration are derived and utilize to convert the fractional differential equations (FDE) into a system of algebraic equations. Finally, some numerical examples are presented to demonstrate the validity, applicability and accuracy of the proposed Bernoulli wavelet method.

Numerical solution of time-delay systems by Hermite wavelet

This paper presents a direct numerical method based on Hermite wavelet to find the solution of time delay systems. The operational matrices of integration, differentiation, production, and delay are derived and utilised to reduce the time-delay dynamical system to a set of algebraic equations. Thus, the problem is simplified to a great extent. The method is easy to implement. The illustrative examples with time-invariant and time-varying coefficients demonstrate the validity of the method.

Hermite wavelet method for solving oscillatory electrical circuit equations

In this research, numerical solutions based on Hermite wavelets have been presented for solving oscillatory electrical circuit equations. The proposed numerical technique is based on Hermite wavelet basis functions, operational matrices of integration and collocation points. Numerical experiments have been performed to illustrate the accuracy and efficiency of the proposed scheme.

Numerical investigation of three-dimensional nanofluid flow with heat and mass transfer on a nonlinearly stretching sheet using the barycentric functions

Purpose The purpose of this paper is to investigate a three-dimensional boundary layer flow with considering heat and mass transfer on a nonlinearly stretching sheet by using a novel operational-matrix-based method. Design/methodology/approach The partial differential equations that governing the problem are converted into the system of nonlinear ordinary differential equations (ODEs) with considering suitable similarity transformations. A direct numerical method based on the operational matrices of integration and product for the linear barycentric rational basic functions is used to solve the nonlinear system of ODEs. Findings Graphical and tabular results are provided to illustrate the effect of various parameters involved in the problem on the velocity profiles, temperature distribution, nanoparticle volume fraction, Nusselt and Sherwood number and skin friction coefficient. Comparison between the obtained results, numerical results based on the Maple's dsolve (type = numeric) command and previous existing results affirms the efficiency and accuracy of the proposed method. Originality/value The motivation of the present study is to provide an effective computational method based on the operational matrices of the barycentric cardinal functions for solving the problem of three-dimensional nanofluid flow with heat and mass transfer. The convergence analysis of the presented scheme is discussed. The benefit of the proposed method (PM) is that, without using any collocation points, the governing equations are converted to the system of algebraic equations.