Operational Matrices Research Articles

Exact operational matrices for rational Bernstein polynomials and its application for solving MHD problems

In this paper, rational Bernstein polynomials on the semi‐infinite interval are adapted to solve a magnetohydrodynamic (MHD) problem. Also, the derivative, product, convert, and Galerkin exact operational matrices (EOMs) of these polynomials are produced. Using the spectral Galerkin method and the exact operational matrices (EOMs) of rational Bernstein polynomials, we solve the problem with high accuracy and speed, and we estimate an exponential convergence rate in this method. The problem is the flow of MHD micropolar over a moving plate with suction and injection boundary conditions. Comparing the results of the rational Bernstein Galerkin method with operational matrices and without operational matrices shows that the present method is faster than another method. Also, comparing the results of the rational Bernstein Galerkin method and rational Gegenbauer tau method shows that the present method is more accurate than another method.

Fractional Pantograph Delay Equations Solving by the Meshless Methods

This work describes two efficient and useful methods for solving fractional pantograph delay equations (FPDEs) with initial and boundary conditions. These two methods depend mainly on orthogonal polynomials, which are the method of the operational matrix of fractional derivative that depends on Bernstein polynomials and the operational matrix of the fractional derivative with Shifted Legendre polynomials. The basic procedure of this method is to convert the pantograph delay equation to a system of linear equations and by using, the operational matrices we get rid of the integration and differentiation operations, which makes solving the problem easier. The concept of Caputo has been used to describe fractional derivatives. Finally, some numerical examples are identified to show the utility and capability of the two proposed approaches. Mathematica®12 program has been relied upon in the calculations.

An efficient method for the fractional electric circuits based on Fibonacci wavelet

In this article, we provide effective computational algorithms based on Fibonacci wavelet (FW) to approximate the solution of fractional order electrical circuits (ECs). The proposed computational algorithm is novel and has not been previously utilized for solving ECs problems. Firstly, we have constructed the operational matrices of fractional integration (OMFI). Secondly, we transform the given initial value problems into algebraic equations, we used the Riemann–Liouville (R–L) fractional integral operator. The proposed approach is capable of handling a wide range of fractional order dynamics in ECs. To validate the effectiveness of the method, four models of electrical circuits with fractional order parameter are considered. The numerical results are compared with exact solutions and absolute errors are calculated to demonstrate the accuracy and efficiency of the approach. The proposed method provides a valuable tool for analyzing and designing fractional order systems in electrical engineering, offering improved accuracy and capturing the intricate behavior of complex systems.

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

PurposeThe purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.Design/methodology/approachThe authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.FindingsThe authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.Originality/valueSince psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

Shifted Chebyshev operational matrices to solve the fractional time-delay diffusion equation

In this paper, Chebyshev operational matrices collocation technique is proposed for solution of variable order derivative within the fractional time-delay diffusion equation. The beginning of this approach is based on the construction of the solution using the shifted Chebyshev polynomials with unknown coefficients. After that, we performed the Newton–Cotes nodal points, the Chebyshev polynomials operational matrices, and the collocation method for calculating the unknown coefficients. According to the described technique, we get an algebraic system of nonlinear equations which can be solved easily by using Newton’s iterative method. The efficiency and applicability of suggested approach are illustrated by some tested examples.

A computational approach for a system of coupled distributed-order fractional Klein–Gordon–Schrödinger equations

In this study, a system of coupled distributed-order fractional Klein–Gordon–Schrödinger equations is introduced. The distributed-order fractional derivative is generated based on the Caputo fractional differentiation. The discrete Chebyshev polynomials are handled to construct computational approach for this system. To do this, fractional derivatives matrices (distribute-order and classical) of the expressed discrete polynomials are obtained. The intended approach is based on approximating the imaginary and real parts of the system solution by the mentioned polynomials and applying the extracted operational matrices, along with employing the collocation technique. The constructed scheme transforms the solution of the original system into solving a related algebraic system. Three test problems are examined to confirm the adequacy of the expressed collocation procedure.

An efficient new numerical algorithm for solving Emden–Fowler pantograph differential equation using Laguerre polynomials

This paper proposes a new approach using the collocation technique for the numerical solution of a class of second-order Emden–Fowler pantograph differential equations. In the literature, all the numerical schemes are designed to deal with only the IVPs of the concerned model. This motivates us to extend the work by considering BVPs as well. First, the uniqueness of the solution to the problem is proved for all three sets of initial/boundary conditions, and then numerical algorithms are established. The algorithms are based on operational matrices of the derivatives, which helps in reducing the computational cost to a great extent. This is shown with the help of CPU time for all the illustrative examples. The formulated schemes handle singularity effectively by using suitable collocation points and simplify the mentioned problems into a nonlinear system of equations, which can be solved by any iterative method. Unlike other methods, the present approach requires no linearization, discretization, or perturbation. The computed numerical results confirm that the present method provides better accuracy than the Bessel collocation method (Izadi and Srivastava, 2021) and the Bernstein operational approach (Sriwastav and Barnwal, 2023). The error bound of the proposed method is also analyzed to support the numerical results.

An Effective Approach Based on Generalized Bernstein Basis Functions for the System of Fourth-Order Initial Value Problems for an Arbitrary Interval

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

In this study, a new hybrid method is developed to solve linear or nonlinear systems of fractional differential equations using Bernoulli wavelets (Bws) and the least squares support vector regression (LS‐SVR). The numerical methods based on operational matrices for solving various kinds of fractional equations have been widely studied in the last decade. In contrast to the existing methods, here we derive the operator of fractional integration, aiming to remove the approximation error. For this purpose, we present Bw operator of Riemann–Liouville fractional integration and use it in our scheme. In the proposed technique, we approximate the unknown functions via Bws, and then with the help of Bw operator of fractional integration and LS‐SVR, we reduce the problem to an algebraic system. In this way, we simplify the computation of the considered system. The error analysis of our method is proposed. Finally, we demonstrate the applicability of the present scheme by solving several numerical examples.

Highly Accurate Method for Boundary Value Problems with Robin Boundary Conditions

The main aim of the current paper is to construct a numerical algorithm for the numerical solutions of second-order linear and nonlinear differential equations subject to Robin boundary conditions. A basis function in terms of the shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions is constructed. It has established operational matrices for derivatives of the constructed polynomials. The obtained solutions are spectral and are consequences of the application of collocation method. This method converts the problem governed by their boundary conditions into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. The theoretical convergence and error estimates are discussed. Finally, we support the presented theoretical study by presenting seven examples to ensure the accuracy, efficiency, and applicability of the constructed algorithm. The obtained numerical results are compared with the exact solutions and results from other methods. The method produces highly accurate agreement between the approximate and exact solutions, which are displayed in tables and figures.

The numerical treatment of fractal‐fractional 2D optimal control problems by Müntz–Legendre polynomials

AbstractIn this work, we introduce a method based on the Müntz–Legendre polynomials (M‐LPs) for solving fractal‐fractional 2D optimal control problems that the fractal‐fractional derivative is described in Atangana‐Riemann‐Liouville's sense. First, we obtain operational matrices of fractal‐fractional‐order derivative, integer‐order integration, and derivative of the M‐LPs. Second, the under study problem is converted into an equivalent variational problem. Then, by applying the M‐LPs, their operational matrices and Gauss–Legendre integration, the mentioned problem is converted to a system of algebraic equations. Finally, this system is solved by Newton's iterative method. Also, we introduce an error bound for the described method. Two examples are included to test the applicability and validity of the present scheme.

A collocation method using generalized Laguerre polynomials for solving nonlinear optimal control problems governed by integro-differential equations

The Laguerre polynomials represent a new set of basic functions adopted to solve a class of optimal control problems (OCP) specified by integro-differential equation namely IOCP. The corresponding operational matrices of derivatives are computed to extend the solution of the problem in terms of Laguerre polynomials. Using the results of the considered basis functions with a joint application of collocation method, the IOCP is reduced to solve a system of nonlinear algebraic system. Come acrossing an error bound and convergence analysis for the approximate optimal value of the performance index have been proved by the proposed method. Finally, some examples are given to bring about the validity and applicability of this technique.

Numerical solution of distributed-order fractional 2D optimal control problems using the Bernstein polynomials

In this work, a new class of two-dimensional optimal control problems is introduced with the help of distributed-order fractional derivative in the Caputo form. The orthonormal Bernstein polynomials are used to make a numerical method to solve these problems. Through this way, some operational matrices are obtained for the classical and fractional derivatives of these polynomials to make effective utilisation of them in constructing the proposed method. The main advantage of the established method is that it turns the solution of the problem under consideration into a system of algebraic equations by approximating the state and control variables by the expressed polynomials and applying the method of Lagrange multipliers. The accuracy and capability of the proposed approach are investigated by solving some numerical examples.

Analytical and Numerical Methods for Solving Second-Order Two-Dimensional Symmetric Sequential Fractional Integro-Differential Equations

In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and L2 errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of 10−14, while the L2 error is on the order of 10−13. Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one.

Estimation of roll damping parameters using Hermite wavelets: An operational matrix of derivative approach

The steady-state ship rolling motion in random beam seas with nonlinear damping and restoring moments are explored mathematically in this work. The Hermite Wavelet Method (HWM) is applied to estimate the parameters in ship roll models. We get the closed-form solutions for the roll angle, damping, and restoring moments using the HWM. Ship motion modeling based on fractional derivatives is also analyzed. The required solution of the governing equation is derived using the operational matrices of Hermite wavelet derivatives. Using operational derivative matrices with the appropriate collocation point, the nonlinear differential equations with boundary conditions are transformed into a system of algebraic equations. The proposed HWM solutions are compared with the results obtained from HPM and OHAM. For small parameter values k and M, the HWM results are much closer to the HPM, OHAM, and numerical MATLAB solutions. Moreover, using HWM is found to be a straightforward, flexible and simple method for solving nonlinear differential equations.

Integro-differential equations: Numerical solution by a new operational matrix based on fourth-order hat functions

This article aims to present a new approach based on the operational matrix method for solving integro-differential equations using numerical techniques. In this method, we employ fourth-degree hat functions (FDHFs) to construct operational matrices. The approach involves two main steps. First, we utilize FDHFs to create operational matrices, which allows us to transform the given problem into a system of algebraic equations. The second step involves solving these algebraic equations numerically. Additionally, we provide an analysis of the errors involved and compare the proposed method with existing techniques. The results demonstrate that the proposed method outperforms its counterparts, highlighting its superiority.

Hybrid Fibonacci wavelet method to solve fractional‐order logistic growth model

The aim of this study is to develop the Fibonacci wavelet method together with the quasi‐linearization technique to solve the fractional‐order logistic growth model. The block‐pulse functions are employed to construct the operational matrices of fractional‐order integration. The fractional derivative is described in the Caputo sense. The present time‐fractional population growth model is converted into a set of nonlinear algebraic equations using the proposed generated matrices. Making use of the quasi‐linearization technique, the underlying equations are then changed to a set of linear equations. Numerical simulations are conducted to show the reliability and use of the suggested approach when contrasted with methods from the existing literature. A comparison of several numerical techniques from the available literature is presented to show the efficacy and correctness of the suggested approach.

Solution Method for Systems of Nonlinear Fractional Differential Equations Using Third Kind Chebyshev Wavelets

Chebyshev Wavelets of the third kind are proposed in this study to solve nonlinear systems of FDEs. The main goal of the method is to convert the nonlinear FDE into a nonlinear system of algebraic equations that can be easily solved using matrix methods. In order to achieve this, we first generate the operational matrices for the fractional integration using third kind Chebyshev Wavelets and block-pulse functions (BPF) for function approximation. Since the obtained operational matrices are sparse, the obtained numerical method is fast and computationally efficient. The original nonlinear FDE is transformed into a system of algebraic equations in a vector-matrix form using the obtained operational matrices. The collocation points are then used to solve the system of algebraic equations. Numerical results for various examples and comparisons are presented.

Solving fractional Bagley-Torvik equation by fractional order Fibonacci wavelet arising in fluid mechanics

This study introduces a new fractional order Fibonacci wavelet technique proposed for solving the fractional Bagley-Torvik equation (BTE), along with the block pulse functions. To convert the specified initial and boundary value problems into algebraic equations, the Riemann–Liouville (R-L) fractional integral operator is defined, and the operational matrices of fractional integrals (OMFI) are built. This numerical scheme’s performance is evaluated and examined on particular problems to show its proficiency and effectiveness, and other methods that are accessible in the current literature are compared. The numerical results demonstrate that the approach produces extremely precise results and is computationally more decisive than previous methods.

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

AbstractWe study the two‐dimensional stochastic time‐fractional Sine–Gordon equation (2D ST‐FS‐G) and provide a solution for it. We use the new clique polynomial method to obtain a numerical solution to the 2D TFS‐G equation. In this technique, the clique polynomial is considered as a basic function for operational matrices. By converting the 2D ST‐FS‐G equation to algebraic equations, a solution is obtained for the desired equation, which clearly shows that this approach is suitable and accurate for dealing with the equation. We further present the error boundary for the obtained approximation of the desired three‐variable function based on the clique polynomial. Finally, we compare some numerical results obtained with the exact ones by a few practical examples.