Legendre Wavelet Galerkin Method Research Articles

Discontinuous Legendre Wavelet Galerkin Method for Solving Lane-Emden Type Equation

Discontinuous Legendre Wavelet Galerkin Method for Solving Lane-Emden Type Equation

Numerical solution of two point boundary value problems by wavelet Galerkin method

In this paper, the Legendre wavelet operational matrix of integration is used to solve two point boundary value problems, in which the coefficients of the ordinary differential equation are real valued functions whose inner product with Legendre wavelet basis functions must exist. The method and convergence analysis of the Legendre wavelet is discussed. This method is applied to solve three boundary value and two moving boundary problems. In boundary value problems, we have studied the effects of condition number, elapse time and relative error on Legendre wavelet. It has been observed that the error decreases as the number of wavelet basis function increases. The condition number of square matrix of matrix equation decreases as Legendre wavelet basis function increases. The Legendre wavelet Galerkin method provides better results in lesser time, in comparison of other methods. In case of moving boundary problems the root mean square error (RMSE) for dimensionless temperature, position of moving interface and its generalized time rate are evaluated. It has been observed that the error increases as Stefan number increases.

AN EXTENDED LEGENDRE WAVELET METHOD FOR SOLVING DIFFERENTIAL EQUATIONS WITH NON-ANALYTIC SOLUTIONS

Although spectral methods such as Galerkin and Tau meth- ods do not work well for solving ordinary differential equations in which, at least, one of the coefficient functions or solution function is not ana- lytic (1), but it is shown that the Legendre wavelet Galerkin method is suitable for solving some kind of these problems (4). In this study we use the extended Legendre wavelet basis and Tau method for solving a wide range of singular boundary value problems. The convergence properties and error analysis of the proposed method are investigated. A compar- ison between the standard Legendre wavelets and extended Legendre wavelets methods shows the capability of the proposed method. AMS Subject Classification: 65Txx; 65L10

Numerical Study on non-Fourier Bio heat Transfer During Thermal Ablation

In this paper, a mathematical model describing the process of non-Fourier heat transfer in biological tissues for generalized coordinate system during thermal ablation by electromagnetic radiation with Gaussian external heat source is studied. The boundary value problem governing this process has been solved using hybrid numerical method taking Legendre wavelet basis function. The result obtained from finite element Legendre wavelet Galerkin method (FELWGM) is compared with exact analytical solution and shows a good agreement. The effect of variability of time, generalized coordinate system, location of tumor or cancer, relaxation time and external heat source coefficient on temperature distribution is discussed in detail. It has been observed that treatment of tumor or cancerous cell is independent of the generalized coordinate system at the thermal ablation position.

Finite Element Legendre Wavelet Galerkin Approch to Inward Solidification in Simple Body Under Most Generalized Boundary Condition

Abstract This paper deals with a mathematical model describing the inward solidification of a melt of phase change material within a container of different geometrical configuration like slab, circular cylinder or sphere under the most generalized boundary conditions. The thermal and physical properties of melt and solid are assumed to be identical. To solve this mathematical model, the finite difference scheme is used to convert the problem into an initial value problem of vector matrix form and further, solving it using the Legendre wavelet Galerkin method. The results thus obtained are analyzed by considering particular cases when one might impose either a constant=time varying temperature or a constant=time varying heat flux or a constant heat transfer coefficient on the surface. The whole analysis is presented in dimensionless form. The effect of variability of shape factor, condition posed at the boundary, Stefan number, Predvoditelev number, Kirpichev number, and Biot number on dimensionless temperature and solid-layer thickness are shown graphically. Furthermore, a comparative study of time for complete solidification is presented.

Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations

By incorporating the Legendre multiwavelet into the mixed discontinuous Galerkin method, in this paper, we present a novel method for solving second-order elliptic partial differential equations (PDEs), which is known as the mixed discontinuous Legendre multiwavelet Galerkin method, derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. The proposed method is also applicable to some other kinds of PDEs.

Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution

In this article, the Legendre wavelet operational matrix of integration is used to solve boundary ordinary differential equations with non-analytic solution. Although the standard Galerkin method using Legendre polynomials does not work well for solving ordinary differential equations in which at least one of the coefficient functions or solution function is not analytic, it is shown that the Legendre wavelet Galerkin method is very efficient and suitable for solving this kind of problems. Several numerical examples are given to illustrate the efficiency and performance of the presented method.