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.     

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.     

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.     

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 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.     

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.      

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.     

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 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.     

Numerical solution of the Riemann-Hilbert problem for a vertical jet under gravity

We consider steady, plane, irrotational and incompressible flow in an infinite channel between two horizontal planes. The flow is formed by two equal but opposite uniform streams approaching from infinity and meeting over a horizontal slot, whence they emerge as a vertical jet under gravity. Conformal mapping is used, and the Riemann-Hilbert solution is obtained for a mixed boundary value problem in the upper half-plane. Numerical methods are introduced, and the problem programmed and run on a digital computer; results are obtained and some difficulties of the computing are discussed.     

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.     

Numerical solution of a turning point problem in elasticity

Summary The axially symmetric equilibria of a stress-free thin elastic spherical cap are studied in dependence of a “thickness” parameter. The attention is focused on the turning points of the corresponding bifurcation diagram. The numerical method is based on an approximation of the osculating parabola in the turning point. The rate of convergence is related to the Fibonacci numbers.