The purpose of this study is to apply the Lie group analysis method to the time-fractional order generalized fifth-order KdV (TFF-KdV) equation. We examine applying symmetry analysis to the TFF-KdV equation with the Riemann–Liouville (R–L) derivative, employing the G ′/ G -expansion approach to yield trigonometric, hyperbolic, and rational function solutions with arbitrary constants. The discovered solutions are unique and have never been studied previously. For solving non-linear fractional partial differential equations, we find that the G ′/ G -expansion approach is highly effective. Finally, conservation laws for the equation are well-built with a full derivation based on the Noether theorem.

An effective numerical method for solving the inverse problem of time fractional parabolic equation is constructed in this paper. We use implicit finite difference method to discretize the problem and for the inverse problem we propose a conjugate gradient type regularization method to solve the discretized ill-posed linear systems. By comparing the different errors and the results with different perturbed data in several numerical experiments, our method is shown to solve the inverse problem, even with some noisy measurements, efficiently and stably.

In this paper, we consider the initial-boundary value problem of determining the stationary right-hand side function in the anomalous diffusion equation with a Caputo fractional derivative with respect to time. The value of the solution of the problem at the final time moment is set as the overdetermination condition. In order to carry out the numerical solution, the iterative conjugate gradient method is used, while at each iteration a direct problem is solved by the finite-difference method using a purely implicit difference scheme. The computational experiment results for the model problem are presented to confirm the efficiency of this new method.

In this paper a numerical scheme based on the idea of radial basis function finite difference (RBF-FD) technique is considered to solve the backward heat conduction problems (BHCP). In the meshless numerical method of RBF-FD, according to the finite difference technique we approximate the required derivatives for every point x(i) is an element of Omega in the corresponding local-support domain Omega(i). Then the partial differential equation problem is transformed into the problem of a linear system of algebraic equations. This method also belongs to localized radial basis function method or the closest point method. To compare RBF-FD method with another RBF technique, radial basis function collocation method (RBFCM) and the method of approximate particular solutions (MAPS) are also considered to solve such inverse problem, and in the computation the standard Tikhonov regularization technique with L-curve method for choose optional regularized parameter is used for solving the highly ill condition system of linear equations. Several numerical examples are presented to demonstrate the ability of the present approach for solving the backward heat conduction problem. (C) 2019 Elsevier Inc. All rights reserved.

In this paper we consider numerical solution of the inverse problem of determining a spacewise dependent right-hand side function, which is also called source function, in parabolic equation from measured data of the solution at the final time point. Such inverse problem occurs in mathematical modeling of various physics or engineering areas. Special iterative method is constructed for the approximate solution of the inverse problem, based on the source function at the new iterative step can be identified by the time derivative and space derivative at final time moment. The capabilities of the proposed computational algorithm is confirmed by the results of several numerical experiments.

Для одномерного параболического уравнения рассмотрена задача определения правой части, зависящей только от пространственных переменных. Для численного решения поставленной обратной начально-краевой задачи используется метод сопряженных градиентов в сочетании с методом конечных разностей с неявной аппроксимацией по времени с весовым множителем $\sigma\in[0, 1]$. Возможности предложенного вычислительного алгоритма подтверждены результатами вычислительного эксперимента для модельных задач с квазиреальными решениями, включая и задачи с условиями переопределения имеющими случайные ошибки. The problem of determining a spacewise-dependent right-hand side in a one-dimensional parabolic equation is considered. The method of conjugate gradients in combination with the method of finite differences with implicit time approximation with the weighting factor $\sigma\in[0, 1]$ is used for the numerical solution of the inverse initial-boundary value problem. The effectiveness of the proposed computational algorithm is confirmed by the results of the computational experiment for model problems with quasi-real solutions, including problems with overdetermination conditions having random errors.

In this article, we propose a new method for the numerical solution of a kind of coefficient inverse problem, in which the time dependent leading coefficient and the right hand side in a parabolic equation need to be identified simultaneously, based on additional information of the solutions at interior points in the computational domain. To solve the nonlinear problem, we use linearized second-order scheme for linearizing the quadratic nonlinear term in time and finite element approximation is used in the space. Based on a special decomposition, at new time level, the original problem is transformed into three standard elliptic problems. The results of numerical experiments are presented to confirm the capability and efficiency of the proposed computational algorithm.

In this paper, we consider the numerical solution of a kind of nonlinear reaction diffusion equations, which describe many important models in the fields of physics, biology and chemistry. A meshless method based on the method of approximate particular solutions (MAPS) and Radial Basis Functions (RBFs) is proposed. Numerical results are compared with the exact solutions and the results of some other proposed methods to confirm the truth of the accuracy and effectiveness of the algorithm. From the numerical examples we can see that the errors are small and the profiles are similar.

Sine-Gordon equation is one of the most famous nonlinear hyperbolic partial differential equations, it arises in many science and engineering fields. In the present work, we consider the numerical solution of two dimensional sine-Gordon equation using localized method of approximate particular solutions (LMAPS), in this technique, the method of approximate particular solutions (MAPS) occurs on some local domains, that greatly reduces the size of the collection matrix, and by combining the conditional positive radial basis function (RBF) generalized thin plate splines (GTPS) with additional low-order polynomial basis to avoid selecting shape parameters during localization. This method is effective compared with other existing methods and since this method is really meshless, it can be used to solve the nonlinear model with complicated computational domains. Several numerical examples are given to demonstrate the ability and accuracy of the present approach for solving nonlinear sine-Gordon equation.

In this paper we consider a numerical method for solving nonhomogeneous backward heat conduction problem. Coupled with the likewise Crank Nicolson scheme and an intermediate variable, the backward problem is transformed to a nonhomogeneous Helmholtz type problem; the unknown initial temperature can be obtained by solving this Helmholtz type problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several problems in both two and three dimensions. The results show that this numerical method can solve nonhomogeneous backward heat conduction problem effectively and precisely, even though the final temperature is disturbed by significant noise.

In this paper, we propose a numerical scheme to solve the inverse problem of determining two lower coefficients that depends on time only in the parabolic equation. The time dependence of the right-hand side of a parabolic equation is determined using additional solution values at points of the computational domain. For solving the nonlinear inverse problem, linearized approximations in time are constructed using the fully implicit scheme, and standard finite difference procedures are used in space. The results of numerical experiments are presented, confirming the capabilities of the proposed computational algorithms for solving the coefficients inverse problem.

We consider a numerical method for solving boundary inverse problem using the implicit difference scheme for approximation by time and finite difference method for the boundary inverse problem. A numerical solution to the boundary inverse problem is determined by special decomposition which transforms the problem into two standard problems. We present the results of numerical experiments, including those with random errors in the input data, which confirm the capabilities of the proposed computational algorithms for solving this boundary inverse problem.

In this paper, we propose a numerical scheme to solve the time-dependent reaction-diffusion equations by using the meshfree method and approximating the solution using multiquadrics (MQ) Radial Basis Function (RBF). The scheme works in very similar fashion as finite difference methods. The results of numerical experiments are presented, and compared with analytical solutions to confirm the good accuracy of the presented scheme.

利用特解方法数值求解一类变系数非齐次电报方程.首先用有限差分方法对时间方向进行离散,进而转化为在空间方向上利用特解方法进行近似,最终利用待定系数法逐步求出在不同时刻的方程的数值解.利用Matlab程序对给定的方程进行数值求解并进行误差估计.通过数值算例,可见所采用的数值方法具有较高的近似精度.

We propose a new numerical meshfree scheme to solve time-dependent problems with variable coefficient governed by telegraph and wave equations which are more suitable than ordinary diffusion equations in modelling reaction diffusion for such branches of sciences. Finite difference method is adopted to deal with time variable and its derivative, and radial basis functions method is developed for spatial discretization. The results of numerical experiments are presented and are compared with analytical solutions to confirm the accuracy of our scheme.

In this paper, we propose a numerical scheme to solve a kind of the nonlinear telegraph equation by using the Kansa’s method with Radial Basis Functions (RBFs). From the numerical results of experiments presented in this paper, we can get that the accuracy between the numerical solutions and the analytical solutions are valid. In this paper, we also give the analysis of the parameter c in IMQ radical basis function for the results.

In this paper,we propose a numerical scheme to solve the one-dimensional(1D)hyperbolic telegraph equation using the Method of Particular Solutions(MPS method)with the Thin Plate Splines(TPS)Radial Basis Function(RBF).The results of numerical experiments are presented,and are compared with analytic solutions to confirm the good accuracy of the presented scheme.