An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.IS

Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options

In this paper we present a robust numerical method to solve several types of European style option pricing problems. The governing equations are described by variants of Black-Scholes partial differential equations (BS-PDEs) of the reaction-diffusion-advection type. To discretise these BS-PDEs numerically, we use the spectral methods in the asset (spatial) direction and couple them with a third-order implicit-explicit predictor-corrector (IMEX-PC) method for the discretisation in the time direction. The use of this high-order time integration scheme sustains the better accuracy of the spectral methods for which they are well-known. Our spectral method consists of a pseudospectral formulation of the BS-PDEs by means of an improved Lagrange formula. On the other hand, in the IMEX-PC methods, we integrate the diffusion terms implicitly whereas the reaction and advection terms are integrated explicitly. Using this combined approach, we first solve the equations for standard European options and then extend this approach to digital options, butterfly spread options, and European calls in the Heston model. Numerical experiments illustrate that our approach is highly accurate and very efficient for pricing financial options such as those described above

A robust spectral method for pricing of American put options on zero-coupon bonds

American put options on a zero-coupon bond problem is reformulated as a linear complementarity problem of the option value and approximated by a nonlinear partial differential equation. The equation is solved by an exponential time differencing method combined with a barycentric Legendre interpolation and the Krylov projection algorithm. Numerical examples shows the stability and good accuracy of the method. A bond is a financial instrument which allows an investor to loan money to an entity (a corporate or governmental) that borrows the funds for a period of time at a fixed interest rate (the coupon) and agrees to pay a fixed amount (the principal) to the investor at maturity. A zero-coupon bond is a bond that makes no periodic interest payments

Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers

Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used

A fitted numerical method for a system of partial delay differential equations

AbstractWe consider a system of two coupled partial delay differential equations (PDDEs) describing the dynamics of two cooperative species. The original system is reduced to a system of ordinary delay differential equations (DDEs) obtained by applying the method of lines. Then we construct a fitted operator finite difference method (FOFDM) to solve this resulting system. The model considered in this paper is very sensitive to small changes in the parameters associated in with the model. Depending on the values of these parameters, the solution can be stable, periodic and/or aperiodic. Such behavior of the solution is exploited via the proposed FOFDM. This FOFDM is analyzed for convergence and it is seen that this method is unconditionally stable and has the accuracy of O(k+h2), where k and h denote time and space step-sizes, respectively. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature

Existence and Permanence in a Diffusive KiSS Model with Robust Numerical Simulations

We have given an extension to the study of Kierstead, Slobodkin, and Skellam (KiSS) model. We present the theoretical results based on the survival and permanence of the species. To guarantee the long-term existence and permanence, the patch size denoted as L must be greater than the critical patch size Lc. It was also observed that the reaction-diffusion problem can be split into two parts: the linear and nonlinear terms. Hence, the use of two classical methods in space and time is permitted. We use spectral method in the area of mathematical community to remove the stiffness associated with the linear or diffusive terms. The resulting system is advanced with a modified exponential time-differencing method whose formulation was based on the fourth-order Runge-Kutta scheme. With high-order method, this extends the one-dimensional work and presents experiments for two-dimensional problem. The complexity of the dynamical model is discussed theoretically and graphically simulated to demonstrate and compare the behavior of the time-dependent density function

Mathematical analysis and numerical simulation of a tumor-host model with chemotherapy application

In this paper, a system of non-linear quasi-parabolic partial differential system, modeling the chemotherapy application of spatial tumor-host interaction is considered. At some certain parameters, we derive the steady state of the anti-angiogenic therapy, baseline therapy and anti-cytotoxic therapy models as well as their local stability condition. We use the method of upper and lower solutions to show that the steady states are globally stable. Since the system of non-linear quasi-parabolic partial differential cannot be solved analytically, we formulate a robust numerical scheme based on the semi-fitted finite difference operator. Analysis of the basic properties of the method shows that it is consistent, stable and convergent. Our numerical results are in agreement with our theoretical findings

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

The objective of this paper is to construct and analyzea fitted operator finite difference method (FOFDM) forthe family of time-dependent singularly perturbed parabolicconvection–diffusion problems. The solution to the problemswe consider exhibits an interior layer due to the presence ofa turning point. We first establish sharp bounds on the solu-tion and its derivatives. Then, we discretize the time variableusing the classical Euler method. This results in a system ofsingularly perturbed interior layer two-point boundary valueproblems. We propose a FOFDM to solve the system above

ON COMPARATIVE GROWTH RELATIONSHIP OF ITERATED ENTIRE FUNCTIONS FROM THE VIEWPOINT OF SLOWLY CHANGING

A positive continuous function L= L(r) is called slowly if L(ar) ~ L(r) as r͢-∞   for every positive constant “a”. Lakshminarasimhan [14] introduced the idea of the functions of L-bounded index. Later Lahiri and Bhattacharjee [16] worked on the entire functions (i.e., functions analytic in the finite complex plane) of L-bounded index and of non uniform L-bounded index. The growth of an entire function f with respect to another entire function g is de.ned as the ratio of their maximum moduli for sufficiently large values of r. The same may be de.ned in terms of maximum terms as well as Nevanlinna’s characteristic functions of entire functions. In this paper we would like to investigate some comparative growth analysis of iterated entire functions (as de.ned by Lahiri and Banerjee [15]) on the basis of their maximum terms, maximum moduli and Nevanlinna.s characteristic functions and obtain some powerful results with a scope of further research in the concerned area

Contour integral method for European options with jumps

We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.Web of Scienc