Numerical treatment for nonlinear Brusselator chemical model

Nowadays, numerical models have great importance in every field of science, especially for solving the nonlinear differential equations, partial differential equations, biochemical reactions, etc. The total time evolution of the reactant species which interacts with other species is simulated by the Runge-Kutta of order four (RK4) and by Non-Standard finite difference (NSFD) method. A NSFD model has been constructed for the biochemical reaction problem and numerical experiments are performed for different values of discretization parameter h. The results are compared with the well-known numerical scheme, i.e. RK4. The developed scheme NSFD gives better results than RK4.     

Numerical solutions of two-dimensional nonlinear integral analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.     

Bifurcation analysis of parametrically excited bipolar disorder model

Bipolar II disorder is characterized by alternating hypomanic and major depressive episode. We model the periodic mood variations of a bipolar II patient with a negatively damped harmonic oscillator. The medications administrated to the patient are modeled via a forcing function that is capable of stabilizing the mood variations and of varying their amplitude. We analyze analytically, using perturbation method, the amplitude and stability of limit cycles and check this analysis with numerical simulations.     

Numerical analysis of nonlinear multiharmonic eddy current problems

Summary This work is devoted to non-linear eddy current problems and their numerical treatment by the so-called multiharmonic approach. Since the sources are usually alternating currents, we propose a truncated Fourier series expansion instead of a costly time-stepping scheme. Moreover, we suggest to introduce some regularization parameter that ensures unique solvability not only in the factor space of divergence-free functions, but also in the whole space H(curl). Finally, we provide a rigorous estimate for the total error that is due to the use of truncated Fourier series, the regularization technique and the spatial finite element discretization.This work has been supported by the Austrian Science Fund Fonds zur Förderung der wissenschaftlichen Forschung (FWF) under the grants SFB F013, P 14953 and START Y192.     

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.     

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.     

Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model

We are interested in the strong convergence of Euler-Maruyama type approximations to the solution of a class of stochastic differential equations models with highly nonlinear coefficients, arising in mathematical finance. Results in this area can be used to justify Monte Carlo simulations for calibration and valuation. The equations that we study include the Ait-Sahalia type model of the spot interest rate, which has a polynomial drift term that blows up at the origin and a diffusion term with superlinear growth. After establishing existence and uniqueness for the solution, we show that an appropriate implicit numerical method preserves positivity and boundedness of moments, and converges strongly to the true solution.     

Stability analysis in a nonlinear ecological model

A high dimension predator-prey model is considered in this paper. Some novel criteria are established for the existence and global asymptotic stability of a unique equilibrium of such model. The approaches are based on fixed point theory, matrix spectral theory and Lyapunov functional. The existence and stability conditions given in terms of spectral radius of explicit matrices are better than conditions obtained by using classic norms. Finally, an example and its simulations show the feasibility of our results.     

Numerical analysis of evolution problems in nonlinear small strains elastoviscoplasticity

Summary The present paper deals with the mathematical and numerical analysis of evolution problems in nonlinear small strains viscoelasticity of Burger's type. After a brief review of the mechanical model, the viscoelastic problem to be solved is written as an abstract evolution problem. The associated operator is proved to be maximal monotone, thus implying existence and uniqueness of solutions. This problem is then solved numerically by a backward Euler discretization in time, a finite element approximation in space and by using a preconditioned conjugate gradient algorithm for solving the resulting nonlinear algebraic systems. Numerical results are finally presented to illustrate the solution procedure.     

Numerical analysis of a modified finite element nonlinear Galerkin method

Summary. A fully discrete modified finite element nonlinear Galerkin method is presented for the two-dimensional equation of Navier-Stokes type. The spatial discretization is based on two finite element spaces X_H and X_h defined on a coarse grid with grid size H and a fine grid with grid size h << H, respectively; the time discretization is based on the Euler explicit scheme with respect to the nonlinear term. We analyze the stability and convergence rate of the method. Comparing with the standard finite element Galerkin method and the nonlinear Galerkin method, this method can admit a larger time step under the same convergence rate of same order. Hence this method can save a large amount of computational time. Finally, we provide some numerical tests on this method, the standard finite element Galerkin method, and the nonlinear Galerkin method, which are in a good agreement with the theoretical analysis.Mathematics Subject Classification (2000): 35Q30, 65M60, 65N30, 76D05     

General decay result for nonlinear viscoelastic equations

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the $L^1(0,\infty )$ relaxation function g namely $g^{\prime }(t)\le ?\xi (t)H(g(t))$, where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when $H(s)=s^p$ and p covers the full admissible range $[1,2)$. We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature.     

Numerical analysis of the hyperbolic two-temperature model

The system of hyperbolic heat conduction problems is solved numerically. The explicit and fully implicit Euler type schemes for the time integration of the nonstationary problem are proposed and investigated. Space derivatives are approximated by using the finite volume method, resulting in conservative and monotonous discrete approximations of the second order of accuracy. The stability analysis is done in the L ₂ and energy norms for a simplified one-temperature equation and the system of two equations, describing the temperature and the flux. Results of numerical experiments are presented. This work was supported by the Lithuanian State Science and Studies Foundation within the projects B-03/2007, B-09/2007 and by the Agency for International Science and Technology Development Programmes in Lithuania within the EUREKA projects E!3691 OPTCABLES and E!3483 EULASNET LASCAN.     

Numerical analysis of a corrected Smagorinsky model

The classical Smagorinsky model's solution is an approximation to a (resolved) mean velocity. Since it is an eddy viscosity model, it cannot represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. This model has recently been corrected to incorporate this flow and still be well-posed. Herein we first develop some basic properties of the corrected model. Next, we perform a complete numerical analysis of two algorithms for its approximation. They are tested and proven to be effective.     

Exponential decay rate of a nonlinear suspension bridge model by a local distributed and boundary dampings

In this paper, we investigate the decay properties of the unconstrained one dimensional suspension bridge model. With only partial damping acting on one or on both equations and with boundary dampings, we prove that the first order energy is decaying exponentially, our method of proof is based on the energy method to build the appropriate Lyapunov functional. Moreover, we develop a numerical algorithm which is based on the finite element method to approximate the spatial variable and the Crank–Nicolson type of symmetric difference scheme to discretize the time derivative, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. At the end, we present some numerical experiments to illustrate our theoretical results.