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This paper is concerned with solving the problem of identifying the control vector problem for a fractional multi-order system of nonlinear ordinary differential equations (ODEs). We describe a quasilinearization approach, based on minimization of a quadratic functional, to compute the values of the unknown parameter vector. Numerical algorithm combining the method with appropriate fractional derivative approximation on graded mesh is applied to SIS and SEIR problems to illustrate the efficiency and accuracy. Tikhonov regularization is implemented to improve the convergence. Results from computations, both with noisy-free and noisy data, are provided and discussed. Simulations with real data are also performed. [ABSTRACT FROM AUTHOR]

A time stepping quasilinearization approach to the mixed (or coupled) form of one and two dimensional Richards' equations is developed. For numerical solution of the linear ordinary differential equation (ODE) for 1D case and elliptic for 2D case, obtained after this semidiscretization, a finite volume method is used for direct problems arising on each time level. Next, we propose a version of the decomposition method for the numerical solution of the inverse ODE and 2D elliptic boundary problems. Computational results for some soil types and its related parameters reported in the literature are presented. [ABSTRACT FROM AUTHOR]

We consider an inverse problem of recovering the mortality rate in the honey bee difference equation model, that tracks a forage honeybee leaving and entering the hive each day. We concentrate our analysis to the model without pesticide contamination in the symmetric spatial environment. Thus, the mathematical problem is formulated as a symmetric inverse problem for reaction coefficient at final time constraint. We use the overspecified information to transform the inverse coefficient problem to the forward problem with non-local terms in the differential operator and the initial condition. First, we apply semidiscretization in space to the new nonsymmetric differential operator. Then, the resulting non-local nonsymmetric system of ordinary differential equations (ODEs) is discretized by three iterative numerical schemes using different time stepping. Results of numerical experiments which compare the efficiency of the numerical schemes are discussed. Results from numerical tests with synthetic and real data are presented and discussed, as well. [ABSTRACT FROM AUTHOR]

In this paper we develop a numerical method for pricing American options under regime-switching model, whose solutions are option buyer indifference prices. The problem is formulated as a nonlinear complementarity problem. We apply interior penalty method to approximate the differential complementarity problem, which results in a system of one degenerate parabolic equation and one ordinary differential equation, weakly coupled by nonlinear exponential term. We formulate the problems in variational form in appropriate spaces and prove comparison principle. Then, as a consequence, we derive estimates about the convergence rate of the interior penalty method. We discretize the penalized problem by fully implicit finite difference scheme and prove that the early exercise constraint is strictly satisfied. We establish comparison principle, uniqueness and boundedness of the numerical solution. To solve the nonlinear system of algebraic equations we use Newton method and partial Newton method. We present numerical results that confirm the theoretical statements. To improve the computational efficiency, we unfold the two-grid idea, combining both iteration processes – Newton method and partial Newton method. We illustrate the efficiency of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, on the basis of Newton’s method, we propose a fast quasilinearization numerical scheme, coupled with Rothe’s method, for fully nonlinear parabolic equations. General conditions that provide quadratic, uniform and monotone convergence of the quasilinearization method (QLM) of solving fully nonlinear ordinary differential equations that arise on each time level, are formulated and elaborated. The convergence of QLM and its rate are examined numerically, on a simple test example with an exact solution. The first few iterations already provide extremely accurate and stable numerical results. The second goal is to consider three applications of the proposed schemes in financial mathematics. Namely, numerical results for three nonlinear problems of optimal investment are presented and discussed. The numerical experiments of the last problem are based on the data from statistic information of the Bulgarian National Bank and Bratislava Interbank. [Copyright &y& Elsevier]