Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation

In this paper we consider the Monte Carlo solution of the Cauchy problem for a nonlinear parabolic equation. Using the fundamental solution of the heat equation, we obtain a nonlinear integral equation with solution the same as the original partial differential equation. On the basis of this integral representation, we construct a probabilistic representation of the solution to our original Cauchy problem. This representation is based on a branching stochastic process that allows one to directly sample the solution to the full nonlinear problem. Along a trajectory of these branching stochastic processes we build an unbiased estimator for the solution of original Cauchy problem. We then provide results of numerical experiments to validate the numerical method and the underlying stochastic representation.     

保守系统中非线性振动问题的数值解法

用目标函数方法寻求保守系统中非线性振动问题的解．以摆的运动作为例子,对相关的微分方程在初位移不为零而初速度为零条件下在时间上进行积分．此时,速度为时间的函数,把此函数称为目标函数．因为摆从右侧到左侧再回到右侧完成一个周期,从而此目标函数的第2个零点便是运动的周期．此外,在数值积分过程中,同时得到了位移函数．此法依赖于常微分方程的数值解法和找函数零点的对分法．某些其它非线性常微分方程的解也得到研究．最后,给出了一些例子和数值结果．     

Numerical solution of a nonlocal identification problem for nonlinear ion transport

A numerical method for the solution of a parameter identification problem in a nonlinear non-self-adjoint two-point boundary value problem with an additional nonlocal condition defining the parameter is presented. The equation arises in the modelling of an experiment known as chronoamperometry for the study of kinetics and mass-transfer in electrochemical events. The algorithm is based on the reformulation of the identification problem as a nonlinear fixed-point problem involving the concentration flux of the reduced species. The linearized boundary value problem is shown to have a unique solution with the unknown parameter uniquely determined by the flux. The linearized BVP is solved using finite differences and the fixed-point is found using the α-bisection method. The results of computational experiments are presented and their physical significance is discussed.     

Nonlinearity problem in the numerical solution of superstiff Cauchy problems

For the numerical solution of Cauchy stiff initial problems, many schemes have been proposed for ordinary differential equation systems. They work well on linear and weakly nonlinear problems. The article presents a study of a number of well-known schemes on nonlinear problems (which include, for example, the problem of chemical kinetics). It is shown that on these problems, the known numerical methods are unreliable. They require a sufficient step reducing at some critical moments, and to determine these moments, sufficiently reliable algorithms have not been developed. It is shown that in the choice of time as an argument, the difficulty is associated with the boundary layer. If the length of the integral curve arc is taken as an argument, difficulties are caused by the transition zone between the boundary layer and regular solution.     

Numerical solution of a third-order nonlinear boundary-value problem by automatic differentiation

We develop a simple numerical method for obtaining Taylor series approximation to the solution of a nonlinear third-order boundary-value problem. We use recursive formulas derived from the governing differential equation itself to calculate exact values of the derivatives needed in the Taylor series. Since we do not use difference formulas or symbolic manipulation for calculating the derivatives, our method requires much less computational effort when compared with the techniques previously reported in the literature. We will illustrate the effectiveness of our method with several test problems.     

Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition

In this paper the singular value decomposition (SVD), truncated at an optimal number, is analysed for obtaining approximate solutions to ill-conditioned linear algebraic systems of equations which arise from the boundary element method (BEM) discretisation of an ill-posed boundary value problem in linear elasticity. The regularisation parameter, namely the optimal truncation number, is chosen according to the discrepancy principle. The numerical results obtained confirm that the SVD+BEM produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data.     

Numerical solution to the Steklov problem

The Steklov problem is considered in a planar domain with a smooth boundary. A numerical algorithm without saturation is constructed. The algorithm allows calculating 3000 eigenvalues with 9 decimal digits.     

Numerical solution of the soil freezing problem

A numerical algorithm is proposed based on the fictitious domain method intended for the mathematical simulation of the freezing of melted soil saturated with aqueous salt solution. The obtained results are compared with those obtained by the technique of front capturing into the grid node and with a self-similar solution.     

Numerical solution of the optimal measurement problem

An algorithm to solve numerically the optimal measurement problem was proposed for the first time. The abstract results were illustrated by numerical experiments.     

A solution of the nonlinear stochastic realization problem

In a recent paper we have constructed a solution of the nonlinear stochastic realization problem. The purpose of this paper is to announce the result to system theorists, as well as to motivate the approach by considering in detail some simple examples.     

Numerical solution of the two-yield elastoplastic minimization problem

Summary This paper concentrates on fast calculation techniques for the two-yield elastoplastic problem, a locally defined, convex but non-smooth minimization problem for unknown plastic-strain increment matrices P ₁ and P ₂. So far, the only applied technique was an alternating minimization, whose convergence is known to be geometrical and global. We show that symmetries can be utilized to obtain a more efficient implementation of the alternating minimization. For the first plastic time-step problem, which describes the initial elastoplastic transition, the exact solution for P ₁ and P ₂ can even be obtained analytically. In the later time-steps used for the computation of the further development of elastoplastic zones in a continuum, an extrapolation technique as well as a Newton-algorithm are proposed. Finally, we present a realistic example for the first plastic and the second time-steps, where the new techniques decrease the computation time significantly.      

An approximate solution for the static beam problem and nonlinear integro-differential equations

We consider a Kirchhoff type nonlinear static beam and an integro-differential convolution type problem, and investigate the effectiveness of the Optimal Homotopy Asymptotic Method (OHAM), in solving nonlinear integro-differential equations. We compare our solutions via the OHAM, with bench mark solutions obtained via a finite element method, to show the accuracy and effectiveness of the OHAM in each of these problems. We show that our solutions are accurate and the OHAM is a stable accurate method for the problems considered.     

Numerical solution of a stability problem in hydrodynamics

Details are given of techniques used for a numerical integration of the Navier-Stokes equations of hydrodynamics in the neighbourhood of a well-known steady solution (the Couette solution in cylindrical coordinates). A rule of numerical stability for the process of successive approximation is proposed; some preliminary results are described. Questo lavoro contiene i primi risultati di una ricerca sulla integrazione numerica di equazioni differenziali alle derivate parziali, iniziata recentemente presso il C. S. C. E. di Pisa. I primi due autori si sono divisi il compito di impostare il problema e di discutere le questioni di stabilità numerica di cui alla Sezione 5; mentre secondo e terzo autore hanno proposto il metodo di soluzione ed hanno condotto gli esperimenti numerici.     

Numerical solution of the kinematic dynamo problem for Beltrami flows in a sphere

The article discusses computational aspects of the kinematic problem of magnetic field generation by a Beltrami flow in a sphere. Galerkin's method is applied with a functional basis consisting of Laplace operator eigenfunctions. Dominant eigenvalues of the magnetic induction operator and associated magnetic eigenmodes are obtained numerically for a certain Beltrami flow for magnetic Reynolds numbers up to 100. The eigenvalue problem is solved by a highly optimized iterative procedure, which is quite general and can be applied to numerical treatment of arbitrary linear stability problems.     

An integral solution to a nonlinear diffusion problem

The generalized Galerkin method (or the method of integral relations) is applied to the type of problem described by quasilinear parabolic equations. As an example the problem of nonlinear transient slab diffusion with a general reservoir boundary condition is worked out. The integral relations are given for an arbitrary number of strips, and solutions using up to seven strips have been obtained in order to investigate the convergence of the method.     

Iterative solution of the nonlinear parabolic periodic boundary value problem

In this paper, the successive over-relaxation method (S.O.R.) is outlined for the numerical solution of the implicit finite difference equations derived from the Crank-Nicolson approximation to a mildly non-linear parabolic partial differential equation with periodic spatial boundary conditions. The usual serial ordering of the equations is shown to be inconsistent, thus invalidating the well known S.O.R. theory of Young (1954), but a functional relationship between the eigenvalues of the S.O.R. operator and the Jacobi operator of a closely related matrix is derived, from which the optimum over-relaxation factor, w_b, can be determined directly. Numerical experiments confirming the theory developed are given for the chosen problem.