arXiv subjects

Kourosh Parand

Publications and source records attributed to Kourosh Parand.

32 records · Page 2Linked to original sources

arXiv2019

Imposing various boundary conditions on positive definite kernels

This work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science. Here we propose a new technique to force the positive definite kernels such as some radial basis functions to satisfy the boundary conditions exactly. It can improve the applications of existing methods based on positive definite kernels and radial basis functions especially the pseudospectral radial basis function method for handling the differential equations with more complicated boundary conditions. The proposed method is applied to a singularly perturbed steady-state convection-diffusion problem, two and three dimensional Poisson's equations with various boundary conditions.

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arXiv2019

A development of Lagrange interpolation, Part I: Theory

In this work, we introduce the new class of functions which can use to solve the nonlinear/linear multi-dimensional differential equations. Based on these functions, a numerical method is provided which is called the Developed Lagrange Interpolation (DLI). For this, firstly, we define the new class of the functions, called the Developed Lagrange Functions (DLFs), which satisfy in the Kronecker Delta at the collocation points. Then, for the DLFs, the first-order derivative operational matrix of $\textbf{D}^{(1)}$ is obtained, and a recurrence relation is provided to compute the high-order derivative operational matrices of $\textbf{D}^{(m)}$, $m\in \mathbb{N}$; that is, we develop the theorem of the derivative operational matrices of the classical Lagrange polynomials for the DLFs and show that the relation of $\textbf{D}^{(m)}=(\textbf{D}^{(1)})^{m}$ for the DLFs is not established and is developable. Finally, we develop the error analysis of the classical Lagrange interpolation for the developed Lagrange interpolation. Finally, for demonstrating the convergence and efficiency of the DLI, some well-known differential equations, which are applicable in applied sciences, have been investigated based upon the various choices of the points of interpolation/collocation.

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arXiv2018

Generalized Lagrangian Jacobi-Gauss-Radau collocation method for solving a nonlinear 2-D optimal control problem with the classical diffusion equation

In this paper, a nonlinear 2D Optimal Control Problem (2DOCP) is considered. The quadratic performance index of a nonlinear cost function is endowed with the state and control functions. In this problem, the dynamic constraint of the system is given by a classical diffusion equation. This article is concerned with a generalization of Lagrangian functions. Besides, a Generalized Lagrangian Jacobi-Gauss-Radau (GLJGR)-collocation method is introduced and applied to solve the aforementioned 2DOCP. Based on initial and boundary conditions, the time and space variables t and x are considered Jacobi-Gauss-Radau points clustered on first or end of interval respectively. Then, to solve the 2DOCP, Lagrange Multipliers are used and the optimal control problem is reduced to a parameter optimization problem. Numerical results demonstrate its accuracy, efficiency, and versatility of the presented method.

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arXiv2017

A Comparison Between Laguerre, Hermite, and Sinc Orthogonal Functions

A series of problems in different fields such as physics and chemistry are modeled by differential equations. Differential equations are divided into partial differential equations and ordinary differential equations which can be linear or nonlinear. One approach to solve those kinds of equations is using orthogonal functions into spectral methods. In this paper, we firstly describe Laguerre, Hermite, and Sinc orthogonal functions. Secondly, we select three interesting problems which are modeled as differential equations over the interval $[0, +\infty)$. Then, we use the collocation method as a spectral method for solving those selected problems and compare the performance of Laguerre, Hermite, and Sinc orthogonal functions in solving those types of equations.

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arXiv2017

A matrix formulation of the Tau method for the numerical solution of non-linear problems

The purpose of this research is to propose a new approach named the shifted Bessel Tau (SBT) method for solving higher-order ordinary differential equations (ODE). The operational matrices of derivative, integral and product of shifted Bessel polynomials on the interval [a, b] are calculated. These matrices together with the Tau method are utilized to reduce the solution of the higher-order ODE to the solution of a system of algebraic equations with unknown Bessel coefficients. The comparisons between the results of the present work and other the numerical method are shown that the present work is computationally simple and highly accurate.

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arXiv2017

Two efficient computational algorithms to solve the singularly perturbed Lane-Emden problem

In this paper, we decide to compare two new approaches based on Rational and Exponential Bessel functions (RBs and EBs) to solve several well-known class of Lane-Emden type models. The problems, which define in some models of non-Newtonian fluid mechanics and mathematical physics, are nonlinear ordinary differential equations of second-order over the semiinfinite interval and have singularity at x = 0. We have converted the nonlinear Lane-Emden equation to a sequence of linear equations by utilizing the quasilinearization method (QLM) and then, these linear equations have been solved by RBs and EBs collocation-spectral methods. Afterward, the obtained results are compared with the solution of other methods for demonstrating the efficiency and applicability of the proposed methods.

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arXiv2016

Numerical Study of Astrophysics Equations by Meshless Collocation Method Based on Compactly Supported Radial Basis Function

In this paper, we propose compactly supported radial basis functions for solving some well- known classes of astrophysics problems categorized as non-linear singular initial ordinary dif- ferential equations on a semi-infinite domain. To increase the convergence rate and to decrease the collocation points, we use the compactly supported radial basis function through the integral operations. Afterwards, some special cases of the equation are presented as test examples to show the reliability of the method. Then we compare the results of this work with some results and show that the new method is efficient and applicable

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arXiv2016

Using Hermite Function for Solving Thomas-Fermi Equation

In this paper, we propose Hermite collocation method for solving Thomas-Fermi equation that is nonlinear ordinary differential equation on semi-infinite interval. This method reduces the solution of this problem to the solution of a system of algebraic equations. We also present the comparison of this work with solution of other methods that shows the present solution is more accurate and faster convergence in this problem.

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arXiv2015

Application of Meshfree Method Based on Compactly Supported Radial Basis Function for Solving Unsteady Isothermal Gas Through a Micro-Nano Porous Medium

In this paper, we have applied the Meshless method based compactly supported radial basis function collocation for obtaining the numerical solution of unsteady gas equation. The unsteady gas equation is a second order non-linear two-point boundary value ordinary differential equation on the semi-infinite domain, with a boundary condition in the infinite. The compactly supported radial basis function collocation method reduces the solution of the equation to the solution of a system of algebraic equation. also, we compare the results of this work with some results. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work

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arXiv2015

RBF DQ Method for Solving Nonlinear Differential Equations of Lane Emden type

The Lane-Emden type equations are employed in the modelling of several phenomena in the areas of mathematical physics and astrophysics . In this paper a new numerical method is applied to investigate some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. We will apply a mesh-less method based on radial basis function differential quadrature method. In RBFs-DQ the derivative value of function with respect to a point is directly approximated by a linear combination of all functional values in the global domain . The main aim of this method is the determination of weight coefficients. Here we concentrate on Gaussian(GS) as a radial function for approximating the solution of the mentioned equation. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.

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arXiv2015

Collocation Method using Compactly Supported Radial Basis Function for Solving Volterra's Population Model

In this paper, indirect collocation approach based on compactly supported radial basis function is applied for solving Volterras population model. The method reduces the solution of this problem to the solution of a system of algebraic equations. Volterras model is a non-linear integro-differential equation where the integral term represents the effect of toxin. To solve the problem, we use the well-known CSRBF: Wendland3,5. Numerical results and residual norm 2 show good accuracy and rate of convergence.

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arXiv2015

Rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems

The Lane-Emden equation has been used to model several phenomenas in theoretical physics, mathematical physics and astrophysics such as the theory of stellar structure. This study is an attempt to utilize the collocation method with the Rational Chebyshev of Second Kind function (RSC) to solve the Lane-Emden equation over the semi-infinit interval [0; +infinity). According to well-known results and comparing with previous methods, it can be said that this method is efficient and applicable.

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arXiv2014

Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options

For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI) based on Wu's compactly supported radial basis functions (WCS-RBFs). LBIE and LRPI schemes are the truly meshless methods, because, a traditional non-overlapping, continuous mesh is not required, either for the construction of the shape functions, or for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the $θ$-weighted scheme is employed for the time derivative. Stability analysis of the methods is analyzed and performed by the matrix method. In fact, based on an analysis carried out in the present paper, the methods are unconditionally stable for implicit Euler (θ= 0) and Crank-Nicolson (θ= 0.5) schemes. It should be noted that LBIE and LRPI schemes lead to banded and sparse system matrices. Therefore, we use a powerful iterative algorithm named the Bi-conjugate gradient stabilized method (BCGSTAB) to get rid of this system. Numerical experiments are presented showing that the LBIE and LRPI approaches are extremely accurate and fast.

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arXiv2014

Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov-Galerkin method

The most recent update of financial option models is American options under stochastic volatility models with jumps in returns (SVJ) and stochastic volatility models with jumps in returns and volatility (SVCJ). To evaluate these options, mesh-based methods are applied in a number of papers but it is well-known that these methods depend strongly on the mesh properties which is the major disadvantage of them. Therefore, we propose the use of the meshless methods to solve the aforementioned options models, especially in this work we select and analyze one scheme of them, named local radial point interpolation (LRPI) based on Wendland's compactly supported radial basis functions (WCS-RBFs) with C6, C4 and C2 smoothness degrees. The LRPI method which is a special type of meshless local Petrov-Galerkin method (MLPG), offers several advantages over the mesh-based methods, nevertheless it has never been applied to option pricing, at least to the very best of our knowledge. These schemes are the truly meshless methods, because, a traditional non-overlapping continuous mesh is not required, neither for the construction of the shape functions, nor for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the implicit-explicit (IMEX) time stepping scheme is employed for the time derivative which allows us to smooth the discontinuities of the options' payoffs. Stability analysis of the method is analyzed and performed. In fact, according to an analysis carried out in the present paper, the proposed method is unconditionally stable. Numerical experiments are presented showing that the proposed approaches are extremely accurate and fast.

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