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

Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration

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

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.

The deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two‐dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well‐posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.

The article gives a brief overview of the possible scope of application of modifications of the Lanchester type models, and also considers some features of the use of simulation elements and numerical solution of differential equations of such models in stochastic formulation. The interest to the models of the Lanchester type, which is traced by the publications in the thematic periodicals, testifies to the effectiveness of such apparatus for solving the problems of operational forecasting of changes in the number of opposing groups during the combat operations. Stochastic formulation allows to simulate in a certain way the influence of random factors and to take into account the elements of uncertainty that influence the dynamics of changes in the number of opposing factions, and which to some extent are present in any combat operations. In contrast to deterministic models, stochastic models require the use of special numerical methods, the choice of a particular model can be justified by the requirements for the degree of their convergence on the integration interval. Evaluation of the convergence can also serve to verify the correctness of the software implementation of the selected method. The article considers a comparative example of using the Milstein method for numerical solution of the Lanchester model differential equations in stochastic formulation.

A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.

Solution of Differential Equations by Using Legendre Wavelet Operational Matrix of Integration

Calculus equation is an important tool for mathematical research and plays an important role in most natural science research. Since the beginning of the eighteenth century, people have gradually used differential and integral equations to solve physical problems. In general, several different aspects of differential equations in the field of mathematics are concerned and studied by most scholars. However, this paper studies and establishes the optimal model for numerical solution of differential equations through deep learning and genetic algorithm. In this paper, the solution of ordinary differential equations is solved through the use of polynomial function space, while the linear combination of simple function x and its product nx can obtain multinomial function space. The space function form of polynomial is very simple, and the operation ability is very strong. Almost all functions can be approximated, and the function space can be transformed by a simple function. Through data simulation test results, it can be found that the oscillation of neural network output is stronger and stronger with the increase in depth, that is to say, the deeper depth endows the neural network with stronger oscillation properties, so for the oscillation function, the depth neural network fitting effect is better than the shallow neural network. Therefore, by combining deep learning and genetic algorithm, this paper studies and establishes the optimal model for numerical solution of differential equations, and finds that the deep neural network can largely complete data simulation.

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.

In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.

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.

A method that employs a dual mesh, one for primary variables and another for dual variables, for the numerical solution of differential equations is presented. The formulation makes use of the traditional finite element interpolation of the primary variables (primal mesh) and the subdomain concept akin to the so-called finite volume method to satisfy the integral form of the governing differential equations on a dual mesh. The approach is termed a dual-mesh finite domain method to distinguish it from both the finite element method and the finite volume method, although the method shares the desirable features of both methods. The method is introduced as applied to representative differential equations in 1-D and 2-D. Numerical examples are presented to illustrate the methodology and accuracy compared to the finite element and finite volume solutions.

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.

Differential equation plays an important role in understanding various phenomenon of science and engineering. Various methods have been developed by mathematicians to obtain a numerical solution of differential equations with minimum errors, transform methods are one of such numerical methods. Till present no transform methods claims to have zero error in obtaining the numerical or exact solution of the given differential equations. Few of well-known transform methods are Laplace transform, Sumudu transforms, Differential transform, Integral transform method etc. In this paper the review of Laplace transform is presented which is one of the tools used by scientists and researchers in finding the solution to their differential equation. In this work, 18 research papers of applications of Laplace transform have been studied and it is shown how this transform has been used by other researchers to obtain the exact solution of ordinary, partial and fractional differential equations. To present a literature review on application of Laplace transform is the main objective of this work.

Algorithms defined by Nash iteration: some implementations via multilevel collocation and smoothing