A continuous Markov process is examined for the purpose of developing Monte Carlo methods for solving partial differential equations. Backward Kolmogorov equations for conditional probability density functions and more general equations satisfied by auxiliary probability density functions are derived. From these equations and the initial and boundary conditions that the density functions satisfy, it is shown that solutions of partial differential equations at an interior point of a region can be written as the expected value of randomly-selected initial and boundary values. From these results, Monte Carlo methods for solving homogeneous and nonhomogeneous elliptic, and homogeneous parabolic partial differential equations are proposed. Hybrid computer techniques for mechanizing the Monte Carlo methods are given. The Markov process is simulated on the analog computer and the digital computer is used to control the analog computer and to form the required averages. Methods for detecting the boundaries of regions using analog function generators and electronic comparators are proposed. Monte Carlo solutions are obtained on a hybrid system consisting of a PACE 231 R-V analog computer and an ALWAC III-E digital computer. The interface for the two computers and a multichannel discrete-interval binary-noise source are described. With this equipment, solutions having a small variance are obtained at a rate of approximately five minutes per solution. Example solutions are given for Laplace's equation in two and three dimensions, Poisson's equation in two dimensions and the heat equation in one, two and three dimensions. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

In the study of many engineering systems involving nonlinear elements such as a saturating inductor in an electrical circuit or a hard spring in a mechanical system, we face the problem of solving the equation ẍ + 2εẋ + x + μx³ = 0 which does not have an exact analytical solution,. Because a consistent framework is desirable in the course of the study, we can assume that the initial conditions are x(0) = 1 and ẋ(0) = 0 without loss of generality. This equation is studied in detail by using numerical solutions obtained from a digital computer. When ε and μ are small, classical methods such as the method of variation of parameters and averaging methods based on residuals provide analytical approximations to the equation and enable the engineer to gain useful insight into the system. However, when ε and μ are not small, these classical methods fail to yield acceptable results because they are all based on the assumption that the equation is quasi-linear. Therefore, two new analytical methods, namely: the parabolic phase approximation and the correction term approximation, are developed according to whether ε < 1 or ε ≥1, and are proven to be applicable for values of ε and μ far beyond the limit of classical methods. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

A necessary and sufficient condition is established for the existence of an invertible mapping of a system of nonlinear differential equations to a system of linear differential equations based on a group analysis of differential equations. It is shown how to construct the mapping, when it exists, from the invariance group of the nonlinear system. It is demonstrated that the hodograph transformation, the Legendre transformation and Lie's transformation of the Monge-Ampere equation are obtained from this theorem. The equation (ux)Puxx-uyy=0 is studied and it is determined for what values of p this equation is transformable to a linear equation by an invertible mapping. Many of the known non-invertible mappings of nonlinear equations to linear equations are shown to be related to invariance groups of equations associated with the given nonlinear equations. A number of such; examples are given, including Burgers' equation uxx +uuz-ut=0 a nonlinear diffusion equation (u⁻²ux ) x -ut =0, equations of wave propagation {Vy-wx=0, Vy-avw-bv-cw=0}, equations of a fluid flow {wy+vx=0, wx -v⁻¹wP=0} and the Liouville equation uxy=eu. As another application of group analysis, it is shown how conservation laws associated with the Korteweg-deVries equation, the cubic Schrodinger equation, the sine-Gordon equation and Hamilton's field equation are related to the invariance groups of the respective equations. All relevant background information is in the thesis, including an appendix on the known algorithm for computing the invariance group of a given system of differential equations. / Science, Faculty of / Mathematics, Department of / Graduate

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