Numerical solution of the two-dimensional Poincaré equation

This paper deals with numerical approximation of the two-dimensional Poincaré equation that arises as a model for internal wave motion in enclosed containers. Inspired by the hyperbolicity of the equation we propose a discretisation particularly suited for this problem, which results in matrices whose size varies linearly with the number of grid points along the coordinate axes. Exact solutions are obtained, defined on a perturbed boundary. Furthermore, the problem is seen to be ill-posed and there is need for a regularisation scheme, which we base on a minimal-energy approach.     

Numerical solution of the Monge–Ampère equation by a Newton's algorithm

We solve numerically the Monge–Ampère equation with periodic boundary condition using a Newton's algorithm. We prove convergence of the algorithm, and present some numerical examples, for which a good approximation is obtained in 10 iterations. To cite this article: G. Loeper, F. Rapetti, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Numerical solution of the sine-Gordon equation

Two new difference schemes are proposed for the sine-Gordon equation. They have the advantage that there is a discrete energy which is conserved. Their stability and convergence is proved. Also we show some numerical computations using one of these schemes.     

Numerical solution of the Burgers’ equation by automatic differentiation

We compute the solution of the one-dimensional Burgers’ equation by marching the solution in time using a Taylor series expansion. Our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non-linear algebraic equations. Instead, we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the Taylor series. We illustrate the effectiveness of our method by solving four test problems with known exact solutions. The numerical solutions we obtain are in excellent agreement with the exact solutions, while being superior to other previously reported numerical solutions.     

Numerical solution to the Kolmogorov-Feller equation

A finite-difference method is proposed for solving the Kolmogorov-Feller integro-differential equation. The numerical scheme constructed is an unconditionally stable marching scheme, and the boundary conditions are determined on the basis of an explicit solution to the original equation at boundary points.     

Numerical solution of one‐dimensional Burgers' equation

In this article, an iterative method for the approximate solution of a class of Burgers' equation is obtained in reproducing kernel space . It is proved the approximation converges uniformly to the exact solution u(x, t) for any initial function under trivial conditions, the derivatives of are also convergent to the derivatives of u(x, t), and the approximate solution is the best approximation under the system © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1251–1264, 2015     

Numerical solution of the abel integral equation

A numerical method for the solution of the Abel integral equation is presented. The known function is approximated by a sum of Chebyshev polynomials. The solution can then be expressed as a sum of generalized hypergeometric functions, which can easily be evaluated, using a simple recurrence relation.     

Numerical solution of the nonlinear Klein-Gordon equation

A numerical method is developed to solve the nonlinear one-dimensional Klein-Gordon equation by using the cubic B-spline collocation method on the uniform mesh points. We solve the problem for both Dirichlet and Neumann boundary conditions. The convergence and stability of the method are proved. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L₂, L_∞ and Root-Mean-Square errors (RMS) in the solutions show the efficiency of the method computationally.     

Numerical solution of Burgers’ equation by the Sobolev gradient method

Burgers’ equation is solved numerically with Sobolev gradient methods. A comparison is shown with other numerical schemes presented in this journal, such as modified Adomian method (MAM) [1] and by a variational method (VM) which is based on the method of discretization in time [2]. It is shown that the Sobolev gradient methods are highly efficient while at the same time retaining the simplicity of steepest descent algorithms.     

Numerical solution of fractional oscillator equation

We focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval. This type of equation includes a complex form of left- and right-sided fractional derivatives. Its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations. Here we elaborated two numerical schemes being dependent on a fractional order of the equation. The results of numerical calculations are compared with analytical solutions. Then we illustrate convergence and stability of our schemes.     

Numerical solution of a secular equation

Summary. A method is proposed for the solution of a secular equation, arising in modified symmetric eigenvalue problems and in several other areas. This equation has singularities which make the application of standard root-finding methods difficult. In order to solve the equation, a class of transformations of variables is considered, which transform the equation into one for which Newton's method converges from any point in a certain given interval. In addition, the form of the transformed equation suggests a convergence accelerating modification of Newton's method. The same ideas are applied to the secant method and numerical results are presented. Received July 1, 1994     

Numerical solution of Volterra integral equation

Summary This paper discusses the use of Gregory's formula for numerical integration of Volterra linear integral equations of the second type. The order of magnitude of the truncation error and the asymptotic behavior of this error are obtained by means of recursive inequalities.     

Numerical solution of the Kolmogorov-Feller equation with singularities

A method is proposed for solving the Kolmogorov-Feller integro-differential equation with kernels containing delta function singularities. The method is based on a decomposition of the solution into regular and singular parts.     

分数阶波方程的数值解法

首先,把分数阶波方程转换成等价的积分-微分方程;然后,利用带权的分数阶矩形公式和紧差分算子分别对时间和空间方向进行离散.证明了当权重为1/2时,时间方向的收敛阶为α,其中α(1α2)为Caputo导数的阶数.利用Gronwall不等式,证明了数值格式的收敛性和稳定性.数值例子进一步表明了数值格式的有效性.     

Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function

This paper presents a numerical method for one-dimensional Burgers’ equation by the Hopf–Cole transformation and a reproducing kernel function, abbreviated as RKF. The numerical solution is given as explicit integral expressions with the RKF at each time step, so that the computation is fully parallel. The stability and error estimates are derived. Numerical results for some test problems are presented and compared with the exact solutions. Some numerical results are also compared with the results obtained by other methods. The present method is easily implemented and effective.     

Numerical solution of a fast diffusion equation

In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: in , a smooth bounded domain in with the zero lateral boundary condition and with a positive initial condition, (fast diffusion problem), and . Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.

     

Numerical solution of a nonlinear integro-differential equation

Galerkin's method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro-differential equation $\text{u}{}_{\mathrm{t}}\text{(x, t) = ∝}{}_0{}^{\mathrm{t}}\text{a(t ? τ)}\text{?}\text{?x}\text{σ(u}{}_{\mathrm{x}}\text{(x, τ)) dτ + f(x, t)}$, 0 < x < 1, 0 < t < T.An error estimate in a suitable norm will be derived for the difference u ? u^h between the exact solution u and the approximant u^h. It turns out that the rate of convergence of u^h to u as h → 0 is optimal. This result was confirmed by the numerical experiments.