1. Fen Fakültesi / Faculty of Science
https://hdl.handle.net/11147/4
2019-10-05T02:09:22ZThe Hirota Method for reaction-diffusion equations with three distinct roots
https://hdl.handle.net/11147/7294
The Hirota Method for reaction-diffusion equations with three distinct roots
Tanoğlu, Gamze; Pashaev, Oktay
The Hirota Method, with modified background is applied to construct exact analytical
solutions of nonlinear reaction-diffusion equations of two types. The first equation has only nonlinear
reaction part, while the second one has in addition the nonlinear transport term. For both cases,
the reaction part has the form of the third order polynomial with three distinct roots. We found analytic
one-soliton solutions and the relationships between three simple roots and the wave speed of
the soliton. For the first case, if one of the roots is the mean value of other two roots, the soliton is
static.We show that the restriction on three distinct roots to obtain moving soliton is removed in the
second case by, adding nonlinear transport term to the first equation.
2004-01-01T00:00:00ZHirota method for solving reaction-diffusion equations with generalized nonlinearity
https://hdl.handle.net/11147/7293
Hirota method for solving reaction-diffusion equations with generalized nonlinearity
Tanoğlu, Gamze
The Hirota Method is applied to find an exact solitary wave solution to evolution
equation with generalized nonlinearity. By introducing the power form of Hirota ansatz the
bilinear representation for this equation is derived and the traveling wave solution is constructed
by Hirota perturbation. We show that velocity of this solution is naturally fixed by truncating
the Hirota’s perturbation expansion. So in our approach, this truncate on works similarly to the
way Ablowitz and Zeppetella obtained an exact travelling wave solution of Fisher’s equation
by finding the special wave speed for which the resulting ODE is of the Painleve type. In the
special case the model admits N shock soliton solution and the reduction to Burgers’ equation.
2006-01-01T00:00:00ZAn operator splitting approximation combined with the supg method for transport equations with nonlinear reaction term
https://hdl.handle.net/11147/7292
An operator splitting approximation combined with the supg method for transport equations with nonlinear reaction term
Baysal, Onur; Tanoğlu, Gamze
In this work, an operator splitting method is proposed in order to obtain
a stable numerical solution for transport equation with non-linear reaction term.
We split the transport equation into a reaction part and an advection diffusion part.
The former one which becomes a nonlinear ordinary differential equation can be
approximated by the simple higher order integrator or solved exactly. The later one
is approximated by the Streamline-Upwind Petrov-Galerkin (SUPG) method combined
with the generalized Euler time integration (q-method). Numerical results
that illustrate the good performance of this method are reported.
2012-01-01T00:00:00ZA conserved linearization approach for solving nonlinear oscillation problems
https://hdl.handle.net/11147/7291
A conserved linearization approach for solving nonlinear oscillation problems
Korkut, Sıla Övgü; Gücüyenen Kaymak, Nurcan; Tanoğlu, Gamze
Nonlinear oscillation problems are extensively used in engineering and applied sciences. Due to non-availability of the analytic solutions, numerical approaches have been used for these equations. In this study, a numerical method which is based on Newton-Raphson linearization and Fréchet derivative is suggested. The convergence analysis is also studied locally. The present method is tested on three examples: damped oscillator, Van-der Pol equation and Schrödinger equation. It is shown that the obtained solutions via the present method are more accurate than those of the well-known second order Runge-Kutta method. When examining the present method, preservation of characteristic properties of these equations is also considered. The obtained results show that the present method is applicable with respect to the efficiency and the physical compatibility.
2018-05-01T00:00:00Z