Browsing by Author "Patidar, Kailash C."
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Item An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection(De Gruyter Open, 2013) Obaid, Hasim; Ouifki, Rachid; Patidar, Kailash C.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.Item Contour integral method for European options with jumps(Elsevier, 2013) Ngounda, Edgard; Patidar, Kailash C.; Pindza, EdsonWe 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.Item Effect of spatial configuration of an extended nonlinear Kierstead-Slobodkin reaction-transport model with adaptive numerical scheme(SpringerOpen, 2016) Owolabi, Kolade M.; Patidar, Kailash C.In this paper, we consider the numerical simulations of an extended nonlinear form of Kierstead-Slobodkin reaction-transport system in one and two dimensions. We employ the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4) schemes proposed by Cox and Matthew (J Comput Phys 176:430-455, 2002), that was modified by Kassam and Trefethen (SIAM J Sci Comput 26:1214-1233, 2005), for the time integration of spatially discretized partial differential equations. We demonstrate the supremacy of ETDRK4 over the existing exponential time differencing integrators that are of standard approaches and provide timings and error comparison. Numerical results obtained in this paper have granted further insight to the question "What is the minimal size of the spatial domain so that the population persists?" posed by Kierstead and Slobodkin (J Mar Res 12:141-147, 1953 ), with a conclusive remark that the popula- tion size increases with the size of the domain. In attempt to examine the biological wave phenomena of the solutions, we present the numerical results in both one- and two-dimensional space, which have interesting ecological implications. Initial data and parameter values were chosen to mimic some existing patternsItem A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer(Wiley, 2019) Munyakazi, Justin B.; Patidar, Kailash C.; Sayi, Mbani T.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.Item A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems(Spring Verlag, 2013) Munyakazi, Justin B.; Patidar, Kailash C.This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings.Item Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options(Kent State University, 2013) Pindza, Edson; Patidar, Kailash C.; Ngounda, EdgardIn 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.Item Mathematical analysis and numerical simulation of a tumor-host model with chemotherapy application(Communications in Mathematical Biology and Neuroscience, 2018) Owolab, Kolade M.; Patidar, Kailash C.; Shikongo, AlbertIn 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.Item Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers(KOREAN MATHEMATICAL SOC, 2014) Munyakazi, Justin B.; Patidar, Kailash C.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.Item A robust spectral method for pricing of American put options on zero-coupon bonds(Global-Science Press, 2018) Pindza, Edson; Patidar, Kailash C.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.Item A robust spectral method for solving Heston’s model(Springer Verlag, 2014) Ngounda, E.; Patidar, Kailash C.; Pindza, E.In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979).