The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Haixiang Zhang - One of the best experts on this subject based on the ideXlab platform.
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crank nicolson quasi wavelets method for solving fourth order Partial Integro differential equation with a weakly singular kernel
Journal of Computational Physics, 2013Co-Authors: Xuehua Yang, Da Xu, Haixiang ZhangAbstract:In this paper, we study a novel numerical scheme for the fourth order Partial Integro-differential equation with a weakly singular kernel. In the time direction, a Crank-Nicolson time-stepping is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term, and the quasi-wavelets numerical method for space discretization. Our interest in the present paper is a continuation of the investigation in Yang et al. [33], where we study discretization in time by using the forward Euler scheme. The comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order Partial Integro-differential equation with a weakly singular kernel. We also tested the method proposed on several one and two dimensional problems with very promising results. Besides, in order to demonstrate the power of the quasi-wavelets method in comparison with standard discretization methods we also consider the high-frequency oscillation problems with the Integro-differential term.
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quintic b spline collocation method for fourth order Partial Integro differential equations with a weakly singular kernel
Applied Mathematics and Computation, 2013Co-Authors: Haixiang Zhang, Xuli Han, Xuehua YangAbstract:Quintic B-spline collocation method for the numerical solution of fourth order Partial Integro-differential equations is considered. The scheme is based on the second order backward differential formula in the time direction and the quintic B-spline method for the spatial derivative. Some numerical experiments are included to demonstrate the effectiveness of the discrete technique. From the numerical experiments, we can find that the present scheme possesses high accuracy and efficiency, and the numerical results are in good agreement with the exact solutions.
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quasi wavelet based numerical method for fourth order Partial Integro differential equations with a weakly singular kernel
International Journal of Computer Mathematics, 2011Co-Authors: Xuehua Yang, Haixiang ZhangAbstract:In this paper, we study the numerical solution of initial boundary-value problem for the fourth-order Partial Integro-differential equations with a weakly singular kernel. We use the forward Euler scheme for time discretization and the quasi-wavelet based numerical method for space discretization. Detailed discrete formulations are given to the treatment of three different boundary conditions, including clamped-type condition, simply supported-type condition and a transversely supported-type condition. Some numerical experiments are included to demonstrate the validity and applicability of the discrete technique. The comparisons of present results with analytical solutions show that the quasi-wavelet based numerical method has a distinctive local property. Especially, the method is easy to implement and produce very accurate results.
Xuehua Yang - One of the best experts on this subject based on the ideXlab platform.
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crank nicolson quasi wavelets method for solving fourth order Partial Integro differential equation with a weakly singular kernel
Journal of Computational Physics, 2013Co-Authors: Xuehua Yang, Da Xu, Haixiang ZhangAbstract:In this paper, we study a novel numerical scheme for the fourth order Partial Integro-differential equation with a weakly singular kernel. In the time direction, a Crank-Nicolson time-stepping is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term, and the quasi-wavelets numerical method for space discretization. Our interest in the present paper is a continuation of the investigation in Yang et al. [33], where we study discretization in time by using the forward Euler scheme. The comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order Partial Integro-differential equation with a weakly singular kernel. We also tested the method proposed on several one and two dimensional problems with very promising results. Besides, in order to demonstrate the power of the quasi-wavelets method in comparison with standard discretization methods we also consider the high-frequency oscillation problems with the Integro-differential term.
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quintic b spline collocation method for fourth order Partial Integro differential equations with a weakly singular kernel
Applied Mathematics and Computation, 2013Co-Authors: Haixiang Zhang, Xuli Han, Xuehua YangAbstract:Quintic B-spline collocation method for the numerical solution of fourth order Partial Integro-differential equations is considered. The scheme is based on the second order backward differential formula in the time direction and the quintic B-spline method for the spatial derivative. Some numerical experiments are included to demonstrate the effectiveness of the discrete technique. From the numerical experiments, we can find that the present scheme possesses high accuracy and efficiency, and the numerical results are in good agreement with the exact solutions.
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quasi wavelet based numerical method for fourth order Partial Integro differential equations with a weakly singular kernel
International Journal of Computer Mathematics, 2011Co-Authors: Xuehua Yang, Haixiang ZhangAbstract:In this paper, we study the numerical solution of initial boundary-value problem for the fourth-order Partial Integro-differential equations with a weakly singular kernel. We use the forward Euler scheme for time discretization and the quasi-wavelet based numerical method for space discretization. Detailed discrete formulations are given to the treatment of three different boundary conditions, including clamped-type condition, simply supported-type condition and a transversely supported-type condition. Some numerical experiments are included to demonstrate the validity and applicability of the discrete technique. The comparisons of present results with analytical solutions show that the quasi-wavelet based numerical method has a distinctive local property. Especially, the method is easy to implement and produce very accurate results.
Jari Toivanen - One of the best experts on this subject based on the ideXlab platform.
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reduced order models for pricing european and american options under stochastic volatility and jump diffusion models
Journal of Computational Science, 2017Co-Authors: Maciej Balajewicz, Jari ToivanenAbstract:Abstract European options can be priced by solving parabolic Partial(-Integro) differential equations under stochastic volatility and jump-diffusion models like the Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a given model parameter variation range.
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reduced order models for pricing european and american options under stochastic volatility and jump diffusion models
2016Co-Authors: Maciej Balajewicz, Jari ToivanenAbstract:European options can be priced by solving parabolic Partial(-Integro) differential equations under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a given model parameter variation range.
Vineet Kumar Singh - One of the best experts on this subject based on the ideXlab platform.
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two dimensional wavelets collocation scheme for linear and nonlinear volterra weakly singular Partial Integro differential equations
International Journal of Applied and Computational Mathematics, 2018Co-Authors: Vijay Kumar Patel, Somveer Singh, Vineet Kumar Singh, Emran TohidiAbstract:In this article, we have study a 2D Legendre and Chebyshev wavelets collocation scheme for solving a class of linear and nonlinear weakly singular Volterra Partial Integro-differential equations (PIDEs). The scheme is based on wavelets collocation for PIDEs with uniquely designed matrices over the Hilbert space defined on the domain $$\left( [0, 1]\times [0, 1]\right) $$ . Using piecewise approximation associated with 2D Legendre wavelet, 2D Chebyshev wavelet and its operational matrices, the considered PIDEs will be reduced into the corresponding system of linear and nonlinear algebraic equations. The corresponding linear and nonlinear system of equations solved by collocation scheme and well-known Newton–Raphson scheme at collocation points respectively. In addition, the convergence and error analysis of the numerical scheme is provided under several mild conditions. The numerical results are correlated with the exact solutions and the execution of the proposed scheme is determined by estimating the maximum absolute errors, $$l_{2}\hbox {-}norm$$ errors and $$l_{\infty }\hbox {-}norm$$ errors. The numerical result shows that the scheme is simply applicable, efficient, powerful and very precisely at small number of basis function. The main important applications of the proposed wavelets collocation scheme is that it can be applied on linear as well as nonlinear problems and can be applied on higher order Partial differential equations too.
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two dimensional shifted legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices
Mathematical Methods in The Applied Sciences, 2017Co-Authors: Vijay Kumar Patel, Somveer Singh, Vineet Kumar SinghAbstract:In this paper, a numerical solution of fractional Partial differential equations (FPDEs) for electromagnetic waves in dielectric media will be discussed. For the solution of FPDEs, we developed a numerical collocation method using an algorithm based on two-dimensional shifted Legendre polynomials approximation, which is proposed for electromagnetic waves in dielectric media. By implementing the Partial Riemann–Liouville fractional derivative operators, two-dimensional shifted Legendre polynomials approximation and its operational matrix along with collocation method are used to convert FPDEs first into weakly singular fractional Partial Integro-differential equations and then converted weakly singular fractional Partial Integro-differential equations into system of algebraic equation. Some results concerning the convergence analysis and error analysis are obtained. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2017 John Wiley & Sons, Ltd.
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numerical solution of nonlinear weakly singular Partial Integro differential equation via operational matrices
Applied Mathematics and Computation, 2017Co-Authors: Somveer Singh, Vijay Kumar Patel, Vineet Kumar Singh, Emran TohidiAbstract:In this paper, we propose and analyze an efficient matrix method based on shifted Legendre polynomials for the solution of non-linear volterra singular Partial Integro-differential equations(PIDEs). The operational matrices of integration, differentiation and product are used to reduce the solution of volterra singular PIDEs to the system of non-linear algebraic equations. Some useful results concerning the convergence and error estimates associated to the suggested scheme are presented. illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.
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operational matrix approach for the solution of Partial Integro differential equation
Applied Mathematics and Computation, 2016Co-Authors: Somveer Singh, Vijay Kumar Patel, Vineet Kumar SinghAbstract:In this paper, an effective numerical method is introduced for the treatment of Volterra singular Partial Integro-differential equations. They are based on the operational and almost operational matrix of integration and differentiation of 2D shifted Legendre polynomials. The methods convert the singular Partial Integro-differential equation in to a system of algebraic equations. Convergence analysis and error estimates are derived for the proposed method. Illustrative examples are included to demonstrate the validity and applicability of the technique.
Z Avazzadeh - One of the best experts on this subject based on the ideXlab platform.
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legendre wavelets for fractional Partial Integro differential viscoelastic equations with weakly singular kernels
European Physical Journal Plus, 2019Co-Authors: Z Avazzadeh, M H Heydari, Carlo CattaniAbstract:This study deals with a new class of fractional Partial Integro-differential equations (FPI-DEs) characterized by the presence of weakly singular kernel and a Newtonian viscoelasticity factor. To numerically solve such equations, a hybrid method is established by combining the Legendre wavelets (LWs), the collocation method, and a new operational matrix of fractional integration (OMFI). More precisely, the unknown solution is expanded by the LWs with unknown coefficients. Then, the OMFI and the collocation method are utilized to extract a system of algebraic equations whose solution is an approximation for the problem’s solution. Convergence and error estimation of the LWs expansion in two dimensions are investigated. Moreover, the efficiency and accuracy of the proposed method are demonstrated by solving some concrete examples. The obtained results confirm the presented approach is very accurate to provide satisfactory solutions.
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smooth solution of Partial Integro differential equations using radial basis functions
Journal of Applied Analysis and Computation, 2014Co-Authors: Z Avazzadeh, M H Heydari, Wen Chen, G B LoghmaniAbstract:In this work, we present the method based on radial basis functions to solve Partial Integro-differential equations. We focus on the parabolic type of Integro-differential equations as the most common forms including the ``\emph{memory}'' of the systems. We propose to apply the collocation scheme using radial basis functions to approximate the solutions of Partial Integro-differential equations. Due to the presented technique, system of linear or nonlinear equations is made instead of primary problem. The method is efficient because the rate of convergence of collocation method based on radial basis functions is exponential. Some numerical examples and investigation of the experimental results show the applicability and accuracy of the method.
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a numerical solution of nonlinear parabolic type volterra Partial Integro differential equations using radial basis functions
Engineering Analysis With Boundary Elements, 2012Co-Authors: Z Avazzadeh, Beygi Z Rizi, F Maalek M Ghaini, G B LoghmaniAbstract:Abstract In this paper, an effective numerical method for solving nonlinear Volterra Partial Integro-differential equations is proposed. These equations include the Partial differentiations of an unknown function and the integral term containing the unknown function which is the “memory” of problem. This method is based on radial basis functions (RBFs) and finite difference method (FDM) which provide the approximate solution. These techniques play the important role to reduce a nonlinear Partial Integro-differential equation to a linear system of equations. Some illustrative examples are shown to describe the method. Numerical examples confirm the validity and efficiency of the presented method.