Numerical solutions of the Kawahara type equations using radial basis functions

2010 ◽  
Vol 28 (2) ◽  
pp. 542-553 ◽  
Author(s):  
Yilmaz Dereli ◽  
Idris Dağ
Keyword(s):  
Radial Basis Functions ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Radial Basis

COMPARATIVE STUDY OF THE MULTIQUADRIC AND THIN-PLATE SPLINE RADIAL BASIS FUNCTIONS FOR THE TRANSIENT-CONVECTIVE DIFFUSION PROBLEMS

2006 ◽  
Vol 17 (08) ◽  
pp. 1151-1169 ◽  
Author(s):  
A. DURMUS ◽  
I. BOZTOSUN ◽  
F. YASUK
Keyword(s):  
Radial Basis Functions ◽  
Thin Plate ◽  
Numerical Solutions ◽  
Convective Diffusion ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Thin Plate Spline ◽  
Collocation Methods ◽  
Radial Basis ◽  
Diffusion Problems

The numerical solutions of the unsteady transient-convective diffusion problems are investigated by using multiquadric (MQ) and thin-plate spline (TPS) radial basis functions (RBFs) based on mesh-free collocation methods with global basis functions. The results of radial basis functions are compared with the mesh-dependent boundary element and finite difference methods as well as the analytical solution for high Péclet numbers. It is reported that for low Péclet numbers, MQ-RBF provides excellent agreement, while for high Péclet numbers, TPS-RBF is better than MQ-RBF.

The meshless method of radial basis functions for the numerical solution of time fractional telegraph equation

2014 ◽  
Vol 24 (8) ◽  
pp. 1636-1659 ◽  
Author(s):  
Akbar Mohebbi ◽  
Mostafa Abbaszadeh ◽  
Mehdi Dehghan
Keyword(s):  
Radial Basis Functions ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Telegraph Equation ◽  
Basis Functions ◽  
Content Type ◽  
Fractional Telegraph Equation ◽  
Radial Basis ◽  
Collocation Approach ◽  
Derivatives Of

Purpose – The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two dimensional time fractional telegraph equation. Design/methodology/approach – In this method the authors first approximate the time fractional derivatives of mentioned equation by two schemes of orders O(τ3−α) and O(τ2−α), 1/2<α<1, then the authors will use the Kansa approach to approximate the spatial derivatives. Findings – The results of numerical experiments are compared with analytical solution, revealing that the obtained numerical solutions have acceptance accuracy. Originality/value – The results show that the meshless method based on the RBFs and collocation approach is also suitable for the treatment of the time fractional telegraph equation.

scholarly journals Use of radial basis functions for meshless numerical solutions applied to financial engineering barrier options

2009 ◽  
Vol 29 (2) ◽  
pp. 419-437 ◽  
Author(s):  
Gisele Tessari Santos ◽  
Maurício Cardoso de Souza ◽  
Mauri Fortes
Keyword(s):  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Analytical Solutions ◽  
Financial Engineering ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Barrier Options ◽  
Black Scholes ◽  
Non Linear ◽  
Radial Basis

A large number of financial engineering problems involve non-linear equations with non-linear or time-dependent boundary conditions. Despite available analytical solutions, many classical and modified forms of the well-known Black-Scholes (BS) equation require fast and accurate numerical solutions. This work introduces the radial basis function (RBF) method as applied to the solution of the BS equation with non-linear boundary conditions, related to path-dependent barrier options. Furthermore, the diffusional method for solving advective-diffusive equations is explored as to its effectiveness to solve BS equations. Cubic and Thin-Plate Spline (TPS) radial basis functions were employed and evaluated as to their effectiveness to solve barrier option problems. The numerical results, when compared against analytical solutions, allow affirming that the RBF method is very accurate and easy to be implemented. When the RBF method is applied, the diffusional method leads to the same results as those obtained from the classical formulation of Black-Scholes equation.

scholarly journals Valuation of Option Pricing with Meshless Radial Basis Functions Approximation

2020 ◽  
pp. 60-70
Author(s):  
M. O. Durojaye ◽  
J. K. Odeyemi
Keyword(s):  
Radial Basis Functions ◽  
Numerical Solutions ◽  
Interpolation Method ◽  
Option Price ◽  
Basis Functions ◽  
Time Stepping ◽  
Runge Kutta Method ◽  
Black Scholes ◽  
Radial Basis ◽  
Marching Method

This work focuses on valuation scheme of European and American options of single asset with meshless radial basis approximations. The prices are governed by Black – Scholes equations. The option price is approximated with three infinitely smooth positive definite radial basis functions (RBFs), namely, Gaussian (GA), Multiquadrics (MQ), Inverse Multiquadrics (IMQ). The RBFs were used for discretizing the space variables while Runge-Kutta method was used as a time-stepping marching method to integrate the resulting systems of differential equations. Numerical examples are shown to illustrate the strength of the method developed. The findings show that the RBFs has proven to be adaptable interpolation method because it does not depend on the locations of the approximation nodes which have overcome frequently evolving problems in computational finance such as slow convergent numerical solutions. Thus, the results allow concluding that the RBF-FD-GA and RBF-FD-MQ methods are well suited for modeling and analyzing Black and Scholes equation.

scholarly journals Numerical Study of Fractional Mathieu Differential Equation Using Radial Basis Functions

2020 ◽  
Vol 7 (4) ◽  
pp. 568-576
Author(s):  
Hojjat Ghorbani ◽  
Yaghoub Mahmoudi ◽  
Farhad Dastmalchi Saei
Keyword(s):  
Differential Equation ◽  
Radial Basis Functions ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Mathieu Equation ◽  
Basis Functions ◽  
Damping Factor ◽  
Radial Basis ◽  
The Right

In this paper, we introduce a method based on Radial Basis Functions (RBFs) for the numerical approximation of Mathieu differential equation with two fractional derivatives in the Caputo sense. For this, we suggest a numerical integration method for approximating the improper integrals with a singularity point at the right end of the integration domain, which appear in the fractional computations. We study numerically the affects of characteristic parameters and damping factor on the behavior of solution for fractional Mathieu differential equation. Some examples are presented to illustrate applicability and accuracy of the proposed method. The fractional derivatives order and the parameters of the Mathieu equation are changed to study the convergency of the numerical solutions.

scholarly journals A Hybrid Approach for the Random Dynamics of Uncertain Systems under Stochastic Loading

2011 ◽  
Vol 2011 ◽  
pp. 1-30 ◽  
Author(s):  
Michele Betti ◽  
Paolo Biagini ◽  
Luca Facchini
Keyword(s):  
Radial Basis Functions ◽  
Numerical Solutions ◽  
Hybrid Approach ◽  
Computational Effort ◽  
Basis Functions ◽  
System Response ◽  
Perturbation Approach ◽  
Stochastic Loading ◽  
Radial Basis

This paper presents a hybrid Galerkin/perturbation approach based on Radial Basis Functions for the dynamic analysis of mechanical systems affected by randomness both in their parameters and loads. In specialized literature various procedures are nowadays available to evaluate the response statistics of such systems, but sometimes a choice has to be made between simpler methods (that could provide unreliable solutions) and more complex methods (where accurate solutions are provided by means of a heavy computational effort). The proposed method combines a Radial Basis Functions (RBF) based Galerkin method with a perturbation approach for the approximation of the system response. In order to keep the number of differential equations to be solved as low as possible, a Karhunen-Loève (KL) expansion for the excitation is used. As case study a non-linear single degree of freedom (SDOF) system with random parameters subjected to a stochastic windtype load is analyzed and discussed in detail; obtained numerical solutions are compared with the results given by Monte Carlo Simulation (MCS) to provide a validation of the proposed approach. The proposed method could be a valid alternative to the classical procedures as it is able to provide satisfactory approximations of the system response.

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