Numerical solution of nonlinear reaction-diffusion and wave equations

Meral, Gülnihal
In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quadrature method (DQM) is used for the spatial discretization of IBVPs and Cauchy problems defined by the nonlinear reaction-diffusion and wave equations. The DRBEM and DQM applications result in first and second order system of ordinary differential equations in time. These systems are solved with three different time integration methods, the finite difference method (FDM), the least squares method (LSM) and the finite element method (FEM) and comparisons among the methods are made. In the FDM a relaxation parameter is used to smooth the solution between the consecutive time levels. It is found that DRBEM+FEM procedure gives better accuracy for the IBVPs defined by nonlinear reaction-diffusion equation. The DRBEM+LSM procedure with exponential and rational radial basis functions is found suitable for exterior wave problem. The same result is also valid when DQM is used for space discretization instead of DRBEM for Cauchy and IBVPs defined by nonlinear reaction-diffusion and wave equations.


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Celebi, AO; Kalantarov, VK (2001-09-01)
Our aim is to establish a spatial decay and growth estimates for solutions of the initial-boundary value problem for the linear wave equation with the damping term under nonlinear boundary conditions.
Effective-mass Klein-Gordon-Yukawa problem for bound and scattering states
Arda, Altug; Sever, Ramazan (2011-09-01)
Bound and scattering state solutions of the effective-mass Klein-Gordon equation are obtained for the Yukawa potential with any angular momentum l. Energy eigenvalues, normalized wave functions, and scattering phase shifts are calculated as well as for the constant mass case. Bound state solutions of the Coulomb potential are also studied as a limiting case. Analytical and numerical results are compared with the ones obtained before. (C) 2011 American Institute of Physics. [doi:10.1063/1.3641246]
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Aydın, Baran; Kanoğlu, Utku (2015-06-22)
We present an initial-boundary value problem formulation for the solution of the nonlinear shallow-water wave (NSW) equations. We transform the nonlinear equations into a linear problem by using the Carrier-Greenspan transformation. Then, we obtain the solution through the separation of variables method rather than integral transform techniques, which is the usual practice (Carrier et al., J Fluid Mech 2003; Kanoglu, J Fluid Mech 2004). This formulation allows the use of any physically realistic initial wav...
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Yıldırım, Raif Orhan (1994-10-01)
Elastic wave propagation through an area discontinuity of two dissimilar, bonded, semi-infinite circular rods is investigated analytically. In particular, the variations of the coefficients of stress reflection and transmission are determined in terms of the nondimensional cross sectional area and mechanical impedance parameters. The coefficients of energy flux reflection and transmission are also included. Then, the case is generalized to include a rigid mass attached at the discontinuity.
Citation Formats
G. Meral, “Numerical solution of nonlinear reaction-diffusion and wave equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2009.