Showing papers on "Finite difference method published in 2021"

Finite-time convergence disturbance rejection control for a flexible Timoshenko manipulator

Abstract: This paper focuses on a new finite-time convergence disturbance rejection control scheme design for a flexible Timoshenko manipulator subject to extraneous disturbances. To suppress the shear deformation and elastic oscillation, position the manipulator in a desired angle, and ensure the finitetime convergence of disturbances, we develop three disturbance observers ( DOs ) and boundary controllers. Under the derived DOs-based control schemes, the controlled system is guaranteed to be uniformly bounded stable and disturbance estimation errors converge to zero in a finite time. In the end, numerical simulations are established by finite difference methods to demonstrate the effectiveness of the devised scheme by selecting appropriate parameters.

Dynamic Optimal Energy Flow in the Heat and Electricity Integrated Energy System

Abstract: To retain intact state information of the district heating network (DHN) as possible in the optimization of heat and electricity integrated energy system (HE-IES), this article combines the transient heat flow and steady-state electric power flow to formulate the dynamic optimal energy flow (OEF) model of HE-IES. For efficient and standardized solution, the finite difference method is applied to convert the partial differential equation constraint (introduced by the temperature dynamics in DHN) into a set of linear equality constraints. The structure of applicable difference schemes for system optimization is analyzed, based on which, a scheme with balanced performances in stability, convergence, simulation accuracy, and computation burden is developed. Moreover, with the proposed multi-objective optimization based method to select proper spatial and temporal calculation step sizes, a compromise between model precision and solution complexity can be reached. Case studies validate the feasibility and benefits of our proposed dynamic OEF computing method, which can further provide support for making optimal planning and operating strategies.

On beta-time fractional biological population model with abundant solitary wave structures

Abstract: The ongoing study deals with various forms of solutions for the biological population model with a novel beta-time derivative operators. This model is very conducive to explain the enlargement of viruses, parasites and diseases. This configuration of the aforesaid classical scheme is scouted for its new solutions especially in soliton shape via two of the well known analytical strategies, namely: the extended Sinh-Gordon equation expansion method (EShGEEM) and the Expa function method. These soliton solutions suggest that these methods have widened the scope for generating solitary waves and other solutions of fractional differential equations. Different types of soliton solutions will be gained such as dark, bright and singular solitons solutions with certain conditions. Furthermore, the obtained results can also be used in describing the biological population model in some better way. The numerical solution for the model is obtained using the finite difference method. The numerical simulations of some selected results are also given through their physical explanations. To the best of our knowledge, No previous literature discussed this model through the application of the EShGEEM and the Expa function method and supported their new obtained results by numerical analysis.

Analysis of Transient Thermal Distribution in a Convective–Radiative Moving Rod Using Two-Dimensional Differential Transform Method with Multivariate Pade Approximant

Abstract: The transient temperature distribution through a convective-radiative moving rod with temperature-dependent internal heat generation and non-linearly varying temperature-dependent thermal conductivity is elaborated in this investigation. Symmetries are intrinsic and fundamental features of the differential equations of mathematical physics. The governing energy equation subjected to corresponding initial and boundary conditions is non-dimensionalized into a non-linear partial differential equation (PDE) with the assistance of relevant non-dimensional terms. Then the resultant non-dimensionalized PDE is solved analytically using the two-dimensional differential transform method (2D DTM) and multivariate Pade approximant. The consequential impact of non-dimensional parameters such as heat generation, radiative, temperature ratio, and conductive parameters on dimensionless transient temperature profiles has been scrutinized through graphical elucidation. Furthermore, these graphs indicate the deviations in transient thermal profile for both finite difference method (FDM) and 2D DTM-multivariate Pade approximant by considering the forced convective and nucleate boiling heat transfer mode. The results reveal that the transient temperature profile of the moving rod upsurges with the change in time, and it improves for heat generation parameter. It enriches for the rise in the magnitude of Peclet number but drops significantly for greater values of the convective-radiative and convective-conductive parameters.

Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and meshfree methods

Abstract: The nonlinear Sine-Gordon equation is one of the widely used partial differential equations that appears in various sciences and engineering. The main purpose of writing this article is providing an efficient numerical method for solving two-dimensional (2D) time-fractional stochastic Sine–Gordon equation on non-rectangular domains. In this method, radial basis functions (RBFs) and finite difference scheme are used to calculate the approximate solution of the mentioned problem. The complexity of solving this problem arises from its high dimension, irregular area, stochastic and fractional terms. Finite difference technique is applied to overcome on the problem dimension, whereas interpolation method based on RBFs is the best idea for solving problems defined in irregular domains. The stochastic Sine–Gordon equation is transformed into elliptic stochastic differential equations using the finite difference method and meshfree method based on RBFs are used to approximate the obtained stochastic differential equation. Some numerical examples are included to investigate the efficiency and accuracy of the presented method.

Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation

Abstract: In this paper, we have investigated some analytical, numerical and approximate-analytical methods by considering the time-fractional nonlinear fractional Burger–Fisher equation (FBFE). (1/G \begin{document}$ ' $\end{document} )-expansion method, finite difference method and Laplace perturbation method have been considered to solve the FBFE. Firstly, we have obtained the analytical solution of the mentioned problem via (1/G \begin{document}$ ' $\end{document} )-expansion method. Also, we have compared the numerical method solutions and have obtained that which method is more effective and accurate. According to the results of this study, it can be concluded that the finite difference method has a lower error level than the Laplace perturbation method. Nonetheless, both of these methods are totally settlement in obtaining efficient results of fractional order differential equations.

Local radial basis function–finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher–Kolmogorov equations

Abstract: In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations.

An efficient local meshless method for the equal width equation in fluid mechanics

Abstract: This paper proposes an accurate and robust meshless approach for the numerical solution of the nonlinear equal width equation. The numerical technique is applied for approximating the spatial variable derivatives of the model based on the localized radial basis function-finite difference (RBF-FD) method. Another implicit technique based on θ − weighted and finite difference methods is also employed for approximating the time variable derivatives. The stability analysis of the approach is demonstrated by employing the Von Neumann approach. Next, six test problems are solved including single solitary wave, fusion of two solitary waves, fusion of three solitary waves, soliton collision, undular bore, and the Maxwellian initial condition. Then, the L 2 and L ∞ norm errors for the first example and the I 1 , I 2 , and I 3 invariants for the other examples are calculated to assess accuracy of the method. Finally, the validity, efficiency and accuracy of the method are compared with those of other techniques in the literature.

The impact of LRBF-FD on the solutions of the nonlinear regularized long wave equation

Abstract: This paper develops a method for the numerical solution of the nonlinear regularized long wave equation This method discretizes the unknown solution in two main schemes The time discretization is accomplished by means of an implicit method based on the $$\theta $$ -weighted and finite difference methods, while the spatial discretization is described with the help of the finite difference scheme derived from the local radial basis function method The advantage of the local collocation method is based only the discretization nodes located in each sub-domain, requiring to be considered when obtaining the approximate solution at every node It also tackles the ill-conditioning problem derived from global collocation method Besides, the stability analysis of the proposed method is analyzed and the accuracy of it is examined with $$L_{\infty }$$ and $$L_2$$ norm errors At the end, the results obtained by the proposed method are compared with the methods given in previous works and it indicates an improvement in comparison with previous works

Finite Difference and Spline Approximation for Solving Fractional Stochastic Advection-Diffusion Equation

Abstract: This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order $$\alpha $$ ( $$0 <\alpha \le 1$$ ). This type of equations due to randomness can rarely be solved, exactly. In this paper, a new approach based on finite difference method and spline approximation is employed to solve time fractional stochastic advection-diffusion type equation, numerically. After implementation of proposed method, the under consideration equation is transformed to a system of second order differential equations with appropriate boundary conditions. Then, using a suitable numerical method such as the backward differentiation formula, the resulting system can be solved. In addition, the error analysis is shown in some mild conditions by ignoring the error terms $$O(\Delta t^2)$$ in the system. In order to show the pertinent features of the suggested algorithm such as accuracy, efficiency and reliability, some test problems are included. Comparison achieved results via proposed scheme in the case of classical stochastic advection-diffusion equation ( $$\alpha =1$$ ) with obtained results via wavelets Galerkin method and obtained results for other values of $$\alpha $$ with the values of exact solution confirm the validity, efficiency and applicability of the proposed method.

A local meshless method to approximate the time-fractional telegraph equation

Abstract: In the present work, we investigate the numerical solution of time-fractional telegraph equation by a local meshless method. The fractional-order derivative is defined in the Caputo’s sense. The time semi-discretization was carried out using finite difference method followed by radial basis function-based spatial discretization. The theoretical convergence analysis and stability analysis of time semi-discrete scheme are also proved. Several test problems with regular and irregular domains with uniform and non-uniform points are considered. To demonstrate the accuracy and efficiency of the proposed method, we compared the analytical and numerical solution of the proposed problem.