Joint with Dr. Mohsen Shahrezaee, Department of Mathematics, Imam Hossein University, Tehran, Iran
In this study, we present a modified configuration, including an exact formulation, for the operational matrix form of the integration, differentiation, and product operators applied in the Galerkin method. Previously, many studies have investigated the methods for obtaining operational matrices (derivative, integral, and product) for Fourier, Chebyshev, Legendre, and Jacobi polynomials, and some have considered the non-orthogonal bases that almost all of them operate on approximately. However, in this study, we aim to obtain the exact operational matrices (EOMs), which can be used for many classes of orthogonal and non-orthogonal polynomials. Similar to previous approaches, this method transforms the original problem into a system of nonlinear algebraic equations. To retain the simplicity of the procedure, the samples are considered in one-dimensional contexts, although the proposed technique can also be employed for two- and three-dimensional problems. Two examples are presented to verify the accuracy of the proposed new approach and to demonstrate the superior performance of EOMs compared with ordinary operational matrices. The corresponding results demonstrate the increased accuracy of the new method. In addition, the convergence of the EOM method is studied numerically and analytically to prove the efficiency of the method.
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.
We propose the use of the meshfree radial basis point interpolation (RBPI) to solve the Black–Scholes model for European and American options. The RBPI meshfree method offers several advantages over the more conventional radial basis function approximation, nevertheless it has never been applied to option pricing, at least to the very best of our knowledge. In this paper the RBPI is combined with several numerical techniques, namely: an exponential change of variables, which allows us to approximate the option prices on their whole spatial domain, a mesh refinement algorithm, which turns out to be very suitable for dealing with the non-smooth options’ payoff, and an implicit Euler Richardson extrapolated scheme, which provides a satisfactory level of time accuracy. Finally, in order to solve the free boundary problem that arises in the case of American options three different approaches are used and compared: the projected successive overrelaxation method (PSOR), the Bermudan approximation, and the penalty approach. Numerical experiments are presented which demonstrate the computational efficiency of the RBPI and the effectiveness of the various techniques employed.
This paper attempts to present a meshless method to find the optimal control of a parabolic distributed parameter system with a quadratic cost functional. The method is based on radial basis functions to approximate the solution of the optimal control problem using collocation method. In this regard, different applications of RBFs are used. To this end, the numerical solutions are obtained without any mesh generation into the domain of the problems. The proposed technique is easy to implement, efficient and yields accurate results. Numerical examples are included and a comparison is made with an existing result.
This work is motivated by studies of numerical simulation for solving the inverse one and two-phase Stefan problem. The aim is devoted to employ two special interpolation techniques to obtain space-time approximate solution for temperature distribution on irregular domains, as well as for the reconstruction of the functions describing the temperature and the heat flux on the fixed boundary x=0 when the position of the moving interface is given as extra specification. The advantage of applying the methods is producing the shape functions which provide the important delta function property to ensure that the essential conditions are fulfilled. Due to ill-posedness of the problem, the process is intractable numerically, so special optimization technique is used to obtain the regularized solution. Numerical results for the typical benchmark test examples, which have the input measured data perturbed by increasing amounts of noise and continuity to the input data in the presence of additive noise, are obtained, which present the efficiency of the proposed method.
In this paper, we state and prove a new formula expressing explicitly the integratives of Bernstein polynomials (or B-polynomials) of any degree and for any fractional-order in terms of B-polynomials themselves. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree-n polynomial on [0,1] into each other. By using their transformation matrices, we derive the operational matrices of integration and product of the Bernstein polynomials. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations.
The Lane–Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. These equations are categorized as non-linear singular ordinary differential equations on the semi-infinite domain . In this research we introduce the Bessel orthogonal functions as new basis for spectral methods and also, present an efficient numerical algorithm based on them and collocation method for solving these well-known equations. We compare the obtained results with other results to verify the accuracy and efficiency of the presented scheme. To obtain the orthogonal Bessel functions we need their roots. We use the algorithm presented by Glaser et al. (SIAM J Sci Comput 29:1420–1438, 2007) to obtain the roots of Bessel functions.
A numerical method for solving optimal control problems is presented in this work. The method is based on radial basis functions (RBFs) to approximate the solution of the optimal control problems by using collocation method. We applied Legendre–Gauss–Lobatto points for RBFs center nodes to use numerical integration method more easily, then the method of Lagrange multipliers is used to obtain the optimum of the problems. For this purpose different applications of RBFs are used. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.
In this paper a novel approach based on the homotopy analysis method (HAM) is presented for solving nonlinear boundary value problems. This method is based on the operational matrix of Chebyshev polynomials to construct the derivative and product of the unknown function in matrix form. In addition, by using the Tau method the problem is converted to a set of algebraic equations from which the solution can be obtained iteratively. The applicability, accuracy and efficiency of this new Tau modification of the HAM is demonstrated via two examples.
Parabolic partial differential equations with an unknown spacewise-dependent coefficient serve as models in many branches of physics and engineering. Recently, much attention has been expended in studying these equations and there has been a considerable mathematical interest in them. In this work, the solution of the one-dimensional parabolic equation is presented by the method proposed by Kansa. The present numerical procedure is based on the product model of the space–time radial basis function (RBF), which was introduced by Myers et al. Using this method, a rapid convergent solution is produced which tends to the exact solution of the problem. The convergence of this scheme is accelerated when we use the Cartesian nodes as center nodes. The accuracy of the method is tested in terms of Error and RMS errors. Also, the stability of the technique is investigated by perturbing the additional specification data by increasing the amounts of random noise. The numerical results obtained show that the proposed method produces a convergent and stable solution.
In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on ℝ and is convergent for any x ∊ℝ. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.
In this research, a new numerical method is applied to investigate the nonlinear controlled Duffing oscillator. This method is based on the radial basis functions (RBFs) to approximate the solution of the optimal control problem by using the collocation method. We apply Legendre–Gauss–Lobatto points for RBFs center nodes in order to use the numerical integration method more easily; then the method of Lagrange multipliers is used to obtain the optimum of the problems. For this purpose different applications of RBFs are used. The differential and integral expressions which arise in the dynamic systems, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.
The present study aims to introduce a solution for parabolic integro-differential equations arising in heat conduction in materials with memory, which naturally occur in many applications. Two Radial basis functions (RBFs) collocation schemes are employed for solving this equation. The first method tested is an unsymmetric method, and the second one, which appears to be more efficient, is a symmetric one. The convergence of these two schemes is accelerated, as we use the cartesian nodes as the center nodes.
In this paper, the problem of solving the two-dimensional diffusion equation subject to a non-local condition involving a double integral in a rectangular region is considered. The solution of this type of problems are complicated. Therefore, a simple meshless method using the radial basis functions is constructed for the non-local boundary value problem with Neumann’s boundary conditions. Numerical examples are included to demonstrate the reliability and efficiency of this method. Also Ne and root mean square errors are obtained to show the convergence of the method.
This paper aims to construct a general formulation for the d-dimensional orthogonal functions and their derivative and product matrices. These matrices together with the Tau method are utilized to reduce the solution of partial differential equations (PDEs) to the solution of a system of algebraic equations. The proposed method is applied to solve homogeneous and inhomogeneous two-dimensional parabolic equations. Also, the mentioned method is employed to find the solution of the coupled Burgers equation. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.
In this paper two numerical meshless methods for solving the Fokker–Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker–Planck equation by using collocation method are applied. The first is based on the Kansa’s approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and Ne errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa’s approach.
In this paper, we apply the Exp-function method to find some exact solutions for two nonlinear partial differential equations (NPDE) and a nonlinear ordinary differential equation (NODE), namely, Cahn-Hilliard equation, Allen-Cahn equation and Steady-State equation, respectively. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving NPDE’s and NODE’s. Mainly we try to present an application of Exp-function method taking to consideration rectifying a commonly occurring errors during some of recent works. The results of the other methods clearly indicate the reliability and efficiency of the used method.
A numerical technique based on the spectral method is presented for the solution of nonlinear Volterra–Fredholm–Hammerstein integral equations. This method is a combination of collocation method and radial basis functions (RBFs) with the differentiation process (DRBF), using zeros of the shifted Legendre polynomial as the collocation points. Different applications of RBFs are used for this purpose. The integral involved in the formulation of the problems are approximated based on Legendre–Gauss–Lobatto integration rule. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme.
In this paper two common collocation approaches based on radial basis functions (RBFs) have been considered; one is computed through the differentiation process (DRBF) and the other one is computed through the integration process (IRBF). We investigate these two approaches on the Volterra’s Population Model which is an integro-differential equation without converting it to an ordinary differential equation. To solve the problem, we use four well-known radial basis functions: Multiquadrics (MQ), Inverse multiquadrics (IMQ), Gaussian (GA) and Hyperbolic secant (sech) which is a newborn RBF. Numerical results and residual norm (‖R(t)‖2) show good accuracy and rate of convergence of two common approaches.
In this study, flow of a third-grade non-Newtonian fluid in a porous half space has been considered. This problem is a nonlinear two-point boundary value problem (BVP) on semi-infinite interval. We find the simple solutions by using collocation points over the almost whole domain [0;∞). Our method based on radial basis functions (RBFs) which are positive definite functions. We applied this method through the integration process on the infinity boundary value and simply satisfy this condition by Gaussian, inverse quadric, and secant hyperbolic RBFs.We compare the results with solution of other methods.
In this paper, we study a non-linear two-point boundary value problem (BVP) on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation (ODE) is investigated by means of the Hermite functions collocation method and the Homotopy analysis method (HAM). The Hermite functions collocation method reduces the solution of above-mentioned problem to the solution of a system of algebraic equations and finds its the numerical solution. The homotopy analysis method is also one of the most effective methods in obtaining series solutions for these types of problems and finds their analytic solution. Through the convergence of these methods we determine the accurate initial slope y′(0) with good capturing the essential behavior of y(x). Numerical and analytical evaluations and comparisons with the results obtained are also discussed at the last part of the paper.