A numerical solution for a stochastic beam equation exhibiting purely viscous behavior

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.

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.

The Taylor series is introduced to analyse and approximate the solution of a class of scaled systems. Using the operational matrix of integration, together with the derived scaled operational matrix, the dynamical equation of a scaled system is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the Taylor series expansion can be determined recursively by the derived algorithm. The parameter identification problems of scaled systems are also investigated. Owing to the simplicity of the operational matrix of integration and the scaled operational matrix, the Taylor series presents considerable computational advantages when compared with the other possible polynomial series, provided that both the input and output signals are analytic functions of t.

An operational matrix for the integration of Chebyshev vector whose elements are Chebyshev polynomials is first derived in this paper. Using the operational matrix of integration, a time‐invariant linear system can be represented by a set of simultaneous linear algebraic equations with the approximation of time functions by Chebyshev series. Analysis and parameter identification of linear system can thus be made by solving the algebraic equations.

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.

A numerical technique is presented for the solution of system of Fredholm integro-differential equations. The method consists of expanding the required approximate solution as the elements of Alpert multiwavelet functions (see Alpert B. et al. J. Comput. Phys. 2002, vol. 182, pp. 149–190). Using the operational matrix of integration and wavelet transform matrix, we reduce the problem to a set of algebraic equations. This system is large. We use thresholding to obtain a new sparse system; consequently, GMRES method is used to solve this new system. Numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results.

We study Legendre polynomials and develop new operational matrix of integration. Based on the operational matrix, we develop a new method to solve a coupled system of Fredholm integral equations of the form U(x)+λ11∫01K11(x,t)U(t)dt+λ12∫01K12(x,t)V(t)dt=f(x), V(x)+λ21∫01K21(x,t)U(t)dt+λ22∫01K22(x,t)V(t)dt=g(x), where λ11, λ12, λ21, and λ22 are real constants and f,g∈C([0,1]). The method reduces the coupled system to a system of easily solvable algebraic equations without discretizing the original system. As an application, we provide examples and numerical simulations demonstrating that the results obtained using the new technique match very well with the exact solutions of the problems. To show the efficiency of the method, we compare our results with some of the results already studied with other available methods in the literature.

General Lagrange-hybrid functions and numerical solution of differential equations containing piecewise constant delays with bibliometric analysis

General orthogonal polynomials are introduced to analyze and approximate the solution of a class of scaled systems. Using the operational matrix of integration, together with the operational matrix of linear transformation, the dynamical equation of a scaled system is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the general orthogonal polynomials can be determined recursively by the derived algorithm. An illustrative example is given to demonstrate the validity and applicability of the orthogonal polynomial approximations.

Analysis of linear distributed parameter systems via double Fourier series

The manuscript investigates the numerical approximation of the fractional mathematical model of the financial crime population dynamics by the Gegenbauer wavelet collocation method. The study aims to enhance the accuracy and efficiency of solving the underlying differential equations that describe these phenomena by utilizing the proposed technique. The financial crime model is a nonlinear coupled system of ordinary differential equations. Using the Gegenbauer wavelets, the novel operational matrices of integration are created. A nonlinear system of ordinary differential equations are transformed into a system of algebraic equations using the characteristics of the Gegenbauer wavelet expansions and the operational matrix of integration, which speeds up processing. Then, this system of algebraic equations is solved using the Newton‐iterative technique to find the unknown Gegenbauer coefficients that help to obtain the approximate solution for the system. A numerical illustration is presented to show the efficacy and precision of the approach. The numerical results obtained from the projected approach are compared with the existing methods, such as NDSolve and Runge Kutta methods. These results show that the projected scheme is simple, reliable, and resilient. The findings suggest that this approach can be a powerful tool for researchers and practitioners in the financial sector, aiding in developing crime prevention and intervention strategies. The study concludes with suggestions for future research directions.

A block pulse operational matrix method for solving two-dimensional nonlinear integro-differential equations of fractional order

This study aims to present the modified Chebyshev wavelet collocation method (CWCM) to investigate and obtain the numerical approximation of CO2 emissions from the energy sector utilizing the fractional mathematical model. The need for energy rises as the population grows. Burning fossil fuels produces a significant portion of the world’s energy, which raises the atmospheric concentration of CO2 and causes global warming. The combination of mathematical modeling studies and numerical simulations allows us to understand the CO2 emissions from the energy sector. Our objective is to build an operational matrix of integration (OMI) based on Chebyshev wavelets and use it to the numerical solution of the CO2 emanation from the energy sector model. Furthermore, these OMI are used to simplify fractional differential equations into an algebraic equation system. Lastly, graphic representations are used to visually present the results and provide empirical support for our theoretical conclusions. Because the CWCM method computes only a few terms and avoids rounding data, it produces correct answers for highly nonlinear situations. We compared the outcomes of the RK4 technique, the ND solver, and the devised method. The numerical results show that CWCM solves the emanation of CO2 from the energy sector with remarkable accuracy and efficiency in both mathematical and environment view. Software called Mathematica has been used for numerical implementation and calculations.

Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration

Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions