A unified derivation of operational matrices for integration in systems analysis

This study aims to propose novel Zernike wavelets and a new method based on the operational matrices for solving nonlinear fractional differential equations. First, non-orthogonal Zernike wavelets are introduced using the Zernike polynomials. Then, a new technique based on combining these wavelets with the block pulse functions is presented to derive the operational matrix of fractional integration and to solve nonlinear fractional differential equations. Moreover, an error analysis is conducted by providing required theorems. Besides, the proposed method is employed to solve a nonlinear fractional competition model of breast cancer. Finally, a parametric study is performed to consider the effect of fractional order on the population of healthy, cancer stem, tumour, and immune cells, as well as the excess estrogen.

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. 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.

The article aims to propose the Lerch operational matrix method to solve a stochastic fractional differential equation. In this approach, the Lerch polynomials have been used as a basis function. Then, the product operational matrix, integral operational matrix, stochastic operational matrix, and operational matrix of fractional integral based on the Lerch polynomials have been constructed. The main characteristic of this method is to reduce the stochastic fractional differential equation into a system of algebraic equations by using derived operational matrices and suitable collocation points. Moreover, the convergence and error analysis of the presented method is also discussed in detail. Additionally, the applicability of the proposed technique is also demonstrated by solving some examples. To confirm the accuracy and effectiveness of the suggested technique, a comparison between the results produced by the proposed method and those obtained by other methods has been provided.

In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

The analysis method of Walsh functions (WF) relies on the use of various operational matrices. To overcome the shortcomings of the existing operational matrices, the paper represents new operational matrices for integration, delay and dyadic derivative, respectively.

The present operational matrix method reduces the Riccati differential equation to a system of algebraic equations. The algebraic system has been solved numerically by Tau method. Convergence analysis of the present method has been discussed in this article. Meanwhile, a numerical method is presented for solving Riccati differential equation. There has also been introduced the operational matrices of derivative and product based on hybrid third kind Chebyshev polynomials and block-pulse functions. Moreover, numerical examples have been included to demonstrate the validity and applicability of the technique.

The generalized block-pulse operational matrices are derived as integral operators for operational calculus. In comparison with Walsh tables, the generalized operational matrices are nothing but the block-pulse tables. Further, it is pointed out that the conventional block-pulse operational matrix is a special case of the generalized operational matrices. Also, the generalized operational matrices are preferable to conventional block-pulse operational matrix when a given function is integrated repeatedly. Finally, the inverse Laplace transform of a rational transfer function via the generalized operational matrices is illustrated as an application of operational calculus.

Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion

Fractional-order Legendre–Laguerre functions and their applications in fractional partial differential equations

Purpose The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically. Design/methodology/approach For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time. Findings By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated. Originality/value Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

In this paper, a generalized operational matrix method based on Haar wavelets is proposed to solve the non-steady heat transfer model of fractional order. Contrary to existing operational matrix methods based on orthogonal functions, we formulate the Haar wavelet operational matrices of general order integration without using the block pulse functions. The main characteristic of our method is that it converts the given problem to a system of algebraic equations with unknown coefficients, which significantly accelerates the entire computational process. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods found in the recent literature. The numerical outcomes show that the proposed technique yields exceptionally precise outcomes and is computationally more efficient than the existing ones.

In this paper, by using the collocation method the operational matrix of Caputo fractional derivative was constructed for the cubic B-spline scaling function and wavelets. We apply the operational matrix and the Collocation method to solve linear fractional Volterra integro-differential equations. By using the principal characteristic of this technique, we have the operational matrix, which is applied to solve any linear fractional Volterra integro-differential equation by reducing the time, computer memory occupation and convert to a system of linear equations. Finally, the validity and applicability of the new technique will be shown by some numerical examples and convergence analysis.

This paper uses new fractional integration operational matrices to solve a class of fractional neutral pantograph delay differential equations. A fractional-order function space is constructed where the exact solution lies in, and a set of orthogonal bases are given. Using them, we reduce the fractional delay differential equation to algebraic equations and get the approximate solution. Finally, we give the Legendre operational matrix of fractional integration to solve the equation as an example and show the efficiency of the method.

Based on using block-pulse functions (BPFs) a new approach for computing optimal control law of linear time-invariant / time-varying, time-delay free / time-delay dynamic systems with quadratic performance index is discussed in this paper. The governing delay-differential equations of dynamic systems are converted into linear algebraic equations by using the operational matrices of integration, delay and product of BPFs. Thus, the problem of finding optimal control law is reduced to the problem of solving algebraic equations obtained via the operational matrices. Three examples are included to demonstrate the applicability of the proposed approach.

The current study is presented to develop two approaches and methodologies to find the numerical solution of linear and non-linear initial value problems such as Lane–Emden type equation, Riccati’s equation and Bessel’s equation of order zero based on approximation. The function approximations (scheme-I and scheme-II) are presented to find the numerical solutions of linear and non-linear initial value problems by using Gauss Legendre roots as node points and random node points in the domain [0, 1]. In the scheme-I, the roots of Legendre polynomial are used as node points for Lagrange polynomials and in scheme-II, we have taken random node points in the domain [0, 1] and orthogonalize the resulting Lagrange polynomials using Gram–Schmidt orthogonalization process. Firstly, we have introduced the function approximations by using generating interpolating scaling functions (ISF) and orthonormal Lagrangian basis functions (OLBF) over the space \(L^{2}[0,1]\) then we have constructed the operational matrices of integration and product operational matrices based on newly designed approximations namely ISF and OLBF. These operational matrices convert given linear and non-linear initial value problems into the associated system of algebraic equations. Finally, we have established error bounds (Lemmas 1, 2) of both scheme-I and scheme-II including the function approximations. The efficiency of the proposed schemes has been confirmed with several test examples including numerical stability. So, the schemes are simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort.