This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients.

We consider the reaction-diffusion-advection (RDA) equations resulting in air pollution mod- eling problems. We employ the finite element method (FEM) for solving the RDA equations in two dimensions. Linear triangular finite elements are used in the discretization of problem domains. The instabilities occuring in the solution when the standard Galerkin finite element method is used, in advection or reaction dominated cases, are eliminated by using an adap- tive stabilized finite element method. In transient problems the unconditionally stable Crank- Nicolson scheme is used for the temporal discretization. The stabilization is also applied for reaction or advection dominant case in the time dependent problems. It is found that the stabilization in FEM makes it possible to solve RDA problems for very small diffusivity constants. However, for transient RDA problems, although the stabilization improves the solution for the case of reaction or advection dominance, it is not that pronounced as in the steady problems. Numerical results are presented in terms of graphics for some test steady and unsteady RDA problems. Solution of an air pollution model problem is also provided.

<p>Doktorska disertacija je posvećena re&scaron;avanju Ko&scaron;ijevog problema odabranih klasa frakcionih diferencijalnih jednačina u okviru Kolomboovih prostora uop&scaron;tenih funkcija. U prvom delu disertacije razmatrane su nehomogene evolucione jednačine sa prostorno frakcionim diferencijalnim operatorima reda 0 &lt; &alpha; &lt; 2 i koeficijentima koji zavise od x i t. Ova klasa jednačina je aproksimativno re&scaron;avana, tako &scaron;to je umesto početne jednačine razmatrana aproksimativna jednačina data preko regularizovanih frakcionih izvoda, odnosno, njihovih regularizovanih množitelja. Za re&scaron;avanje smo koristili dobro poznate uop&scaron;tene uniformno neprekidne polugrupe operatora. U drugom delu disertacije aproksimativno su re&scaron;avane nehomogene frakcione evolucione jednačine sa Kaputovim<br />frakcionim izvodom reda 0 &lt; &alpha; &lt; 2, linearnim, zatvorenim i gusto definisanim<br />operatorom na prostoru Soboljeva celobrojnog reda i koeficijentima koji zavise<br />od x. Odgovarajuća aproksimativna jednačina sadrži uop&scaron;teni operator asociran sa polaznim operatorom, dok su re&scaron;enja dobijena primenom, za tu svrhu&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />u disertaciji konstruisanih, uop&scaron;tenih uniformno neprekidnih operatora re&scaron;enja.<br />U oba slučaja ispitivani su uslovi koji obezbeduju egzistenciju i jedinstvenost<br />re&scaron;enja Ko&scaron;ijevog problema na odgovarajućem Kolomboovom prostoru.</p> / <p>Colombeau spaces of generalized functions. In the firs part, we studied inhomogeneous evolution equations with space fractional differential operators of order 0 &lt; &alpha; &lt; 2 and variable coefficients depending on x and t. This class of equations is solved&nbsp; approximately, in such a way that instead of the originate equation we considered the corresponding approximate equation given by regularized fractional derivatives, i.e. their&nbsp; regularized multipliers. In the solving procedure we used a well-known generalized uniformly continuous semigroups of operators. In the second part, we solved approximately inhomogeneous fractional evolution equations with Caputo fractional derivative of order 0 &lt; &alpha; &lt; 2, linear, closed and densely defined operator in Sobolev space of integer order and variable coefficients depending on x. The corresponding approximate equation&nbsp;&nbsp; is a given by the generalized operator associated to the originate&nbsp; operator, while the solutions are obtained by using generalized uniformly continuous solution operators, introduced and developed for that purpose. In both cases, we provided the conditions that ensure the existence and uniqueness solutions of the Cauchy problem in some Colombeau spaces.</p>