# Generalized Numerical Solution of Mixed Systems of Elliptic and Hyperbolic Partial, Differential Equations

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Thesis

In this thesis transformation techniques are used to develop normal forms for systems of elliptic and hyperbolic partial differential equations as well as for mixed systems of same types of equations. 1. A generalized finite difference scheme with defined parameters for the solution of any type of second-order parial differential equation of at most three independent variables is developed and implemented on a Computer. Stability conditions for elliptic, hyperbolic and parabolic types of general second partial differential equations in two independent variables were established; thus completing the work of P.D. Lax and L. Collatz in this area. Results of the solution of Euler's second order partial differential equation, using this scheme, are presented. 2. A generalized solution scheme suitable for the solution of systems of linear first-order elliptic and hyperbolic partial differential equations as well as mixed systems of same types of equations, was developed and implemented on a Computer. Consistency, stability and convergence conditions were developed as functions of the elements of the system coefficient matrix. The scheme was used to solve a variety of systems of first-order partial differential equations. The results are presented. 3. A generalized scheme for the solution of systems of first-order non-linear partial types were derived from (2) above. Conditions of consistency, stability and convergence were also derived. Whilst some of the results obtained in: (1) Above are important and new, those obtained in (2) are not only new but are considered a major contribution to the pool of knowledge on finite-difference methods for the solution of partial differential equations. Often in Mechanical, Electrical, and Aeronautical Engineering as well as in Economic model's design, physical and economic systems are modelled mathematically by systems of partial differential equations. Sometimes for a variety of reasons, such as time ependence and parameter dependence, the elements of the coefficient matrix of the defining system of equations vary to such an extent that the type of the defining system of equations varies e.g from elliptic to hyperbolic (and vice versa) or to mixtures of both. Indeed sometimes only the general form, but not the type, of equation is known, and the system coefficient matrix cannot be continually monitored. The results of this thesis facilitate implementation of adaptive control schemes for such systems.

- Theses1065