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Higher-order finite-difference methods for partial differential equationsCheema, Tasleem Akhter January 1997 (has links)
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.
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The Finite Element Method Solution Of Reaction-diffusion-advection Equations In Air PollutionTurk, Onder 01 September 2008 (has links) (PDF)
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.
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Uopštena rešenja nekih klasa frakcionih parcijalnih diferencijalnih jednačina / Generalized Solutions for Some Classes of Fractional Partial Diferential EquationsJapundžić Miloš 26 December 2016 (has links)
<p>Doktorska disertacija je posvećena rešavanju Košijevog problema odabranih klasa frakcionih diferencijalnih jednačina u okviru Kolomboovih prostora uopštenih funkcija. U prvom delu disertacije razmatrane su nehomogene evolucione jednačine sa prostorno frakcionim diferencijalnim operatorima reda 0 < α < 2 i koeficijentima koji zavise od x i t. Ova klasa jednačina je aproksimativno rešavana, tako što je umesto početne jednačine razmatrana aproksimativna jednačina data preko regularizovanih frakcionih izvoda, odnosno, njihovih regularizovanih množitelja. Za rešavanje smo koristili dobro poznate uopštene uniformno neprekidne polugrupe operatora. U drugom delu disertacije aproksimativno su rešavane nehomogene frakcione evolucione jednačine sa Kaputovim<br />frakcionim izvodom reda 0 < α < 2, linearnim, zatvorenim i gusto definisanim<br />operatorom na prostoru Soboljeva celobrojnog reda i koeficijentima koji zavise<br />od x. Odgovarajuća aproksimativna jednačina sadrži uopšteni operator asociran sa polaznim operatorom, dok su rešenja dobijena primenom, za tu svrhu <br />u disertaciji konstruisanih, uopštenih uniformno neprekidnih operatora rešenja.<br />U oba slučaja ispitivani su uslovi koji obezbeduju egzistenciju i jedinstvenost<br />rešenja Košijevog 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 < α < 2 and variable coefficients depending on x and t. This class of equations is solved 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 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 < α < 2, linear, closed and densely defined operator in Sobolev space of integer order and variable coefficients depending on x. The corresponding approximate equation is a given by the generalized operator associated to the originate 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>
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