Iterative Operator Splitting Method For A Class Of Fisherâs Equation
The class of Fisher’s equations plays a significant role in the modeling of physical situations such as growth model in ecology, fluid dynamics, gas dynamics and heat transfer. In the study of partial differential equation, these equations are categorized as nonlinear parabolic type which describ...
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| Format: | Dissertations |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/36037 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | The class of Fisher’s equations plays a significant role in the modeling of physical
situations such as growth model in ecology, fluid dynamics, gas dynamics and heat
transfer. In the study of partial differential equation, these equations are categorized
as nonlinear parabolic type which describes the evolution of a density or
concentration with respect to time. This thesis is devoted to the study of numerical
solution of class of Fisher’s equation using operator splitting method. In particular,
the iterative operator splitting method is discussed. This method splits a difficult
problem into several sub-problems which may be easier to solve. Each sub-problem
is solved using a combination of an iterative scheme with suitable integrators.
We carried our analysis by considering the theoretical and numerical aspects. In
the theoretical part, consistency, stability, and convergence of the numerical scheme
are discussed. Consistency analysis is done by considering the boundedness of
operators and the choices of initial solutions. Boundedness of operators plays the
main role in deriving the local error by utilizing their nice properties. However,
when the operator is unbounded such nice properties are no longer valid. In order
to derive the local error for the unbounded case, the properties from semigroup
theory are used. The stability analysis is done by considering the stability with
respect to initial condition. We showed that the change of the solutions with respect
to the change of initial conditions are bounded. The convergence analysis are done
using the LadyWindermere’s Fan argument to relate the local and the global errors.
We showed that the convergence of the scheme by showing that the global error is
bounded. In the numerical part, we apply the scheme to Fisher’s equation. The
maximum error is observed and then the errors are compared with non iterative
operator splitting. |
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