Numerical Methods Of Integral(Differential) Equations Based On Moiite-carlo Technique

Monte Carlo (MC) method is also called random sampling technique or statisticaltest method. Over past half a century, with the development of science and technol-ogy and the invention of the computer, this method was proposed as an independentmethod, and then applied to the test and the development of nuclear weapons firstly.Monte Carlo method is a numerical method based on the theory of probability andstatistics. But it is very diferent with the general numerical methods. Monte Carlomethod can describe the characteristics of object and the process of experiment re-alistically, as well as solve some problems which the general numerical methods aredifcult to solve. Hence, the field of application of Monte Carlo method is extensiveincreasingly. Monte Carlo method can be used to solve not only the problems withrandom process, but also the deterministic problems, such as multidimensional inte-grals, inverse matrix, partial diferential equation and integral equation with the initialand bounded conditions, and so on.As a kind of numerical method, Monte Carlo method has a great many merits. Forexample, it has only a few moments that Monte Carlo method is limited to geometricrestrictions, the convergence speed is irrelevant to the dimensions of the problem,the error is easy to determine, the program structure is simple, flexible and easy toaccomplish. On the other hand, it has some drawbacks at the same time, such as aslow convergence speed and a probabilistic error etc.. So, in theory and practice it ismeaningful to combine Monte Carlo method with numerical method for solving thenumerical solutions of equations (integral equations and partial diferential equations)making the most use of its merits. This paper laid the foundation for further detailedresearch of Monte Carlo method application in the future.In this paper, we study the linear Fredholm and Volterra types Integral equationsrelated to soil mechanics and also discuss the nonlinear Fredholm type integral equationand parabolic partial diferential equation about thermodynamics. These equationshave the potential mechanical background. The main work of this dissertation can besummarized as follows:1. Because there is no randomness on integral equation problems, therefore, it is need to construct a reasonable probability model in order to the numerical character-istic just is the solution of integral equations. Based on this idea, in Chapters2and5,at first we use the Simpson quadrature formula and the Taylor series solution methodfor solving integral equations that yields linear system. In the second place Markovrandom process model with discrete state is established and then the mathematicalexpectation of random variable defined by this probability model is the solution ofintegral equations. Monte Carlo method is employed to simulate this established ran-dom process model. In Chapter3, first of all, we use the successive approximationsmethod for solving integral equations that yields iterative term. Secondly, Monte Car-lo method with importance sampling based on the simulation of a continuous randomprocess is employed to solve the solution of this iterative form. In Chapter4, firstly, wetransform the integral equations into integrating problem by the successive approxima-tions method. Secondly, Monte Carlo variance reduction techniques based on modifiedcontrol variate is used to deal with the generated multidimensional integral. The prob-ability error is reduced. In Chapters2,3,4,5, Numerical experiments indicate thatthe proposed Monte Carlo is more flexible and simple. Its program structure design issimple and easy to realize. And it can calculate independently an unknown functionalof the solution, in a given number of operations, at only one point of domain or thelinear combination of an unknown function at several points of domain.2. In Chapter6, Monte Carlo method for solving nonlinear integral equationsis researched. We intend to transform this kind equation into nonlinear algebraicsystem by numerical methods and the optimization problem is generated and thenMonte Carlo random search algorithm is employed to solve this optimization problem.Finally, some examples are given to demonstrate the validity and applicability of theproposed approach.3. In Chapter7, Monte Carlo method is provided to evaluate the temperaturedistribution of a one-dimensional linear parabolic partial diferential equation. Theadvantages of Monte Carlo method are took full advantage of solving the sparse system.Linear algebraic system with sparse coefcient matrix is given by using the compactfinite diference and Markov process model with the discrete state is established. Itis theoretically proved that mathematical expectation of defined random variable bythis probability model is the solution of this kind diferential equation. The computed results of numerical examples with the combined use of the CFD technique and MonteCarlo method are illustrated in order to show the efciency and the accuracy of theMonte Carlo method.In the end, all mentioned above are summarized and some conclusions are putforth.