Solución bidimensional sin malla de la ecuación no lineal de convección-difusión-reacción mediante el método de Interpolación Local Hermítica

A meshless numerical scheme is developed for solving a generic version of the non-linear convection-diffusion-reaction equation in two-dimensional domains. The Local Hermitian Interpolation (LHI) method is employed for the spatial discretization and several strategies are implemented for the solution of the resulting non-linear equation system, among them the Picard iteration, the Newton Raphson method and a truncated version of the Homotopy Analysis Method (HAM). The LHI method is a local collocation strategy in which Radial Basis Functions (RBFs) are employed to build the interpolation function. Unlike the original Kansa’s Method, the LHI is applied locally and the boundary and governing equation differential operators are used to obtain the interpolation function, giving a symmetric and non-singular collocation matrix. Analytical and Numerical Jacobian matrices are tested for the Newton-Raphson method and the derivatives of the governing equation with respect to the homotopy parameter are obtained analytically. The numerical scheme is verified by comparing the obtained results to the one-dimensional Burgers’ and two-dimensional Richards’ analytical solutions. The same results are obtained for all the non-linear solvers tested, but better convergence rates are attained with the Newton Raphson method in a double iteration scheme.Se desarrolla un esquema numérico sin malla para resolver una versión genérica de la ecuación no lineal de convección-difusión-reacción en dominios bidimensionales. El método de Interpolación Hermitiana Local (LHI) se emplea para la discretización espacial y se implementan varias estrategias para la solución del sistema de ecuaciones no lineal resultante, entre ellas la iteración Picard, el método Newton Raphson y una versión truncada del Método de Análisis de Homotopía. (JAMÓN). El método LHI es una estrategia de colocación local en la que se utilizan funciones de base radial (RBF) para construir la función de interpolación. A diferencia del método original de Kansa, el LHI se aplica localmente y los operadores diferenciales de ecuación límite y gobernante se utilizan para obtener la función de interpolación, dando una matriz de colocación simétrica y no singular. Las matrices analíticas y numéricas jacobianas se prueban para el método de Newton-Raphson y las derivadas de la ecuación de gobierno con respecto al parámetro de homotopía se obtienen analíticamente. El esquema numérico se verifica comparando los resultados obtenidos con las soluciones analíticas unidimensionales de Burgers y Richards bidimensionales. Se obtienen los mismos resultados para todos los solucionadores no lineales probados, pero se obtienen mejores tasas de convergencia con el método Newton Raphson en un esquema de doble iteración

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.info:eu-repo/semantics/publishedVersio

On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Method

(R1969) On the Approximation of Eventual Periodicity of Linearized KdV Type Equations using RBF-PS Method

Water wave propagation phenomena still attract the interest of researchers from many areas and with various objectives. The dispersive equations, including a large body of classes, are widely used models for a great number of problems in the fields of physics, chemistry and biology. For instance, the Korteweg-de Vries (KdV) equation is one of the famous dispersive wave equation appeared in the theories of shallow water waves with the assumption of small wave-amplitude and large wave length, also its various modifications serve as the modeling equations in several physical problems. Another interesting qualitative characteristic of solutions of some dispersive wave equations indicated through experiments that are connected with their large-time behavior termed as Eventual Time Periodicity which is exhibited by solutions of initial-boundary-value problems (IBVPs henceforth). Laboratory experiments in a channel with a flap-type or piston-type wave maker mounted at one end of a channel exposed this interesting phenomena. Here in this study we numerically investigate the solutions periodicity for linearized KdV type equations on a finite (bounded) domain with periodic boundary conditions using meshfree technique known as Radial basis function pseudo spectral (RBF-PS) method

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

The aim of this paper is to introduce a new numerical method for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation. This method is combination of group preserving scheme (GPS) with radial basis functions (RBFs), which takes advantage of two powerful methods, one as geometric numerical integration method and the other meshless method. Thus, we introduce this method as the Lie-group method based on radial basis functions (LG–RBFs). In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. One of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions. To demonstrate this point, we solve nonlinear GBBMB equation on various geometric domains in one, two and three dimension spaces. The results of numerical experiments are compared with analytical solutions and the method presented in Dehghan et al. (2014) to confirm the accuracy and efficiency of the presented method

Solving burger’s equantion using explicit finite difference method and method of line

Burgers’ equation is a quasilinear differential equation can be solve either analytically or numerically. The analytical solutions use the Hopf-Cole transformation and reduced to diffusion equation. The focus of this research was to solve Burgers’ equation numerically by using Finite Difference Method (FDM) and Method of Line (MOL) by using Fourth Order Runge-Kutta (RK4). The accuracy of MOL obtained solutions depends on the type of Ordinary Differential Equation (ODE) method used. The results obtained from both numerical method were compared between Hopf-Cole transformation analytical solutions. The simulations is coded by using MATLAB software. From the comparison, both methods shown to be good numerical approximation as the results obtained near to the exact solution. As the increase of spatial step size, the solutions obtained with be more accurate followed by individual methods’ restrictions. Different time and viscosity coefficient also tested to observe the changes of Burgers’ equation solutions