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In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d-dimensional problem in space into d+1-dimension...
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In this paper, the approximate solution u(x, t) of the one-dimensional fractional Rayleigh-Stokes equation with initial and boundary conditions has been studied. This equation has the main part in defining the dynamic character of some non-Newtonian fluids. Time discretization is performed by new finite methods of the first order, and the RBF metho...
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... In all these works, they made difference between space and time variables and discussed the time stability analysis. In this paper the numerical scheme developed by Hamaidi et al. in [1][2][3] , called the space-time localized RBF collocation method is applied to solve the Black-Scholes equations governing European and American options. The technique is based on the transformation of the -dimensional (PDE) into -dimensional one by combining the -dimensional vector space variable and -dimensional time variable in one -dimensional variable vector and solving the problem without using any time discretization techniques like implicit, explicit, the method-of-line approach and others as done by classical methods. ...
... Where and are the risk-free interest rate and the volatility of the stock price respectively. Under some known assumptions, the option value satisfies the one-dimensional Black-Scholes equation defined by (2) Before illustrating how to apply the radial basis functions as a spatial-temporal collocation scheme for options pricing, we first give a review of the two options. ...
... First, we consider the one-dimensional European options problem, which satisfies the Black-Scholes equation given by (2). To solve numerically this problem, we consider a truncated domain . ...
In this paper, we investigate the use of space-time localized radial basis function collocation method to solve the options pricing European and American models. The application of the proposed method to options pricing European and American models open a new area in the development of this technique for solving partial differential equations (PDEs) with variable coefficients. Beside the known advantages of the method when solving the PDEs with constant coefficients, re-computing the resulting matrix and solving the algebraic system at each time level, as done by time stepping method when solving PDEs with variable coefficients, are avoided. In herein, the numerical approximation of the optimal exercise boundary in the case of American options is obtained effectively by using a new algorithm of penalty iterative scheme. The same technique is applied to the two and three-dimensional American options without boundary conditions when considering the problem in the space domain. Results are compared with analytical and available published numerical results. Obtained results indicate that the proposed scheme offers accurate approximation compared with existing numerical methods applied to such option pricing European and American models. 1. Introduction Solving partial differential equations arising in financial markets is nowday considered as one of the most important problems that attract the attention of both mathematicians and scientists. Many techniques have been developed and applied to solve European and American options. Finite element method, boundary element method and finite differential method are the most famous used mesh methods. All of them are based on seeking the approximate solution at each time level using time discretization scheme. Recently, some papers treating the multi-asset American put option were published. Among them, we can mention the work based on a semi-implicit finite element scheme published by R. Zhang et al. [5]. They have used a penalty method to transform the linear complementary problem (LCP) defining the multi-asset American put option into a nonlinear parabolic problem on an unbounded domain. For solving the problem, they implemented the far field boundary condition to determinate a rectangular truncated domain after variables change. The mentioned mesh methods show difficulties when it has to be applied to option pricing problem with number of multi-asset great than four. During the last decade and under the intensive research on meshless methods, many papers [6-18] have been published concerning the application of radial basis function approach as meshfree method for solving PDEs in financial option pricing. In many of these works, equations governing European and American options were firstly time discretized by using time integration schemes and then, the obtained option pricing equations in space were solved at each time level by using radial basis function meshless method. To complete the use of such meshless method and because of the use of time integration scheme, time stability analysis issue has to be discussed. In [15] two schemes of a meshless local weak form of the boundary element method (LBEM) for option pricing are presented. The first one is based on the moving least squares approximation (MLS) and the second on Wu's compactly supported radial basis functions. The problem of the free boundary arising in American option is reduced to a problem with an unbounded fixed boundary using a Richardson extrapolation technique. The problem is then solved in a truncated domain , where is chosen five time of the strike price. The-method is used for time discretization and the approximate solution is obtained at every time step level by solving a sparse system of linear equations. In [16] J.A. Rad et al. had also used the radial basis point interpolation (RBPI) to solve the Black-Scholes model for European and American options. To overcome the
... However, these traditional space discrete methods usually combine with the time discrete method for solving transient problems. To improve the computation efficiency, several space-time coupled methods are proposed by researchers [24][25][26][27]. By introducing the space-time domain and space-time radial basis function (STRBF) [28,29], the space-time backward substitution method (STBSM) was proposed for solving transient advection-diffusion reaction equations in irregular domain [24,30]. ...
... A simple engineering component is considered in the last example with temperature boundary conditions. As shown in Fig. 13a and Fig. 13b, the outside diameter is 2.5, the inside diameter is 1.5, the diameter of side holes is 0. 25 and thermal conductivity coefficient (x) are given by [10] Table 6 compares the computational accuracy and efficiency between STBSM and traditional BSM combined with Crank-Nicolson scheme at different time step = 0.1 s, 0.01 s, 0.001 s. As the results listed in Table 6, the advantage of STBSM is that it takes less time to obtain the results which are similar to traditional BSM. ...
In this paper, an efficient space-time backward substitution method (STBSM) is presented for solving nonlinear transient heat conduction problems in 2D and 3D functionally graded materials (FGMs). To improve the computation efficiency of the traditional backward substitution method (BSM) for transient problems, the space-time radial basis function (STRBF) and space-time trigonometric basis function (STTBF) are introduced for interpolation in the proposed method. The time dimension is regarded as a pseudo-space dimension to generate the space-time Euler metric rT in STRBF, which can be applied to transform the d-dimensional transient governing equation into (d+1)-dimensional steady-state governing equation. Benefiting from merits of the traditional BSM, an accurate numerical solution is available with only a few of computational points. Several examples are solved to verify the accuracy and efficiency of the proposed method and its merits and flaws are discussed in the end.
... Many numerical methods presented for approximation solution of algebraic equations in (Hafiz, 2013;Wahls and Poor, 2015;Cordero et al., 2010;Parhi and Gupta, 2008.;Ham et al., 2008;Proinov and Ivanov, 2015;Proinov, 2016;Binwal, 2021;Hamaidi et al., 2021;Kodnyanko, 2021;Tassaddiq et al., 2021). ...
In the present study numerical solution for non linear equations have been studied. Ridders methodd have been discussed. An improvement of Ridders method with combination of Bisection and newton Raphson methods have been presented. An algorithm for the proposed method have been stated. Moreover, several examples are included to demonstrate the validity and applicability of the presented technique. Matlab program involved for numerical computations. The proposed method applied for given examples. The error analysis table presents the obtained numerical results. The numerical solutions which found by Matlab program has good results in terms of accuracy.
... For example, fourth-order PDEs are used to model floor systems, bridge slabs, aviation wings, and window glasses [24,25]. A lot of problems in the fields of electrochemistry, electromagnetics, material science, diffusion processes, and chaotic dynamics can be modeled by FDEs [26][27][28]. The parabolic PDEs play a vital role in the study of viscoelastic and inelastic flows, beam deformation, and layer deflection [29,30]. ...
In this article, natural transform iterative method
has been used to find the approximate solution of fractional
order parabolic partial differential equations of multidimensions
together with initial and boundary conditions.
The method is applicable without any discretization or
linearization. Three problems have been taken as test
examples and the results are summarized through plots
and tables to show the efficiency and reliability of the
method. By practice of a few iterations, we observe that
the approximate solution of the parabolic equations converges
to the exact solution. The fractional derivatives are
considered in the Caputo’s sense.
... Li et al. [19] used the global space-time radial basis function for the estimation of river pollution source. Hamaidi et al. [20,21] proposed a space-time localized RBF collocation method for solving parabolic and hyperbolic equations and the authors show the advantages of using the techniques combing the local method and spacetime techniques, in terms of stability and computational cost as there is no need to recompute the matrix of algebraic system at each time level. ...
... In this study, we proposed efficient computational techniques based on the mesh-free local radial basis function collocation method and space-time techniques [20,21,[29][30][31][32][33] , to numerically solve the Richards equation. The Gardner model [8] is used for the unsaturated hydraulic conductivity function. ...
... In this section, we introduce the concept of the space-time meshfree method. Such technique has been analyzed and applied in previous studies for several problems [20,21,[29][30][31][32][33] . The technique is based on the radial basis function collocation method [11,12] which has recently become very popular due to its advantages in terms of approximation properties of solutions and its less computational cost since it does not require mesh generation on the considered computational domain. ...
In this study, we focus on space-time mesh-free numerical techniques for efficiently solving the Richards equation which is often used to model unsaturated flow through porous media. We propose an efficient approach which combines the use of local multiquadric (MQ) radial basis function (RBF) methods and space-time techniques. The localized MQ-RBFs meshless methods allow to avoid mesh generation and ill-conditioning problem where a sparse matrix is obtained for the global system which has the advantage of using reduced memory and computational time. To further reduce the computational cost, we use the space-time techniques having the advantages of solving the resulting algebraic system only once and removing the time-integration procedure. The proposed method has the benefit of considering collocation points on the boundaries of computational domains which makes it more flexible in dealing with complex geometries. We implement the proposed numerical model of infiltration and we perform a series of numerical tests, encompassing various nontrivial solutions, to confirm the performance of the proposed techniques. The numerical simulations show the accuracy, efficiency in terms of computational cost, and capability of the proposed numerical techniques in solving the Richards equation in two-, three- and four-dimensional space-time domains with complex boundaries.
In this paper, the space-time generalized finite difference method (ST-GFDM) with supplementary nodes is applied to solve the thin elastic plate bending under dynamic loading. The proposed method treats the temporal dimension as another spatial dimension, thus the d-dimensional problem in space can be viewed as a new (d+1)-dimensional problem in the space-time domain. This method effectively reduces the complexity of numerical solution of the problem by simultaneously discretizing the temporal and spatial dimensions in the space-time domain. It avoids the temporal-difference measurement while maintaining all the advantages of the GFDM. The ST-GFDM leads a sparse linear system due to its local meshless feature, which is suitable for solving large-scale and long-time bending problems. Moreover, the supplementary nodes on the boundary are introduced to make the system well-determined. Several numerical examples are illustrated to demonstrate the accuracy and stability of the proposed method.
