Solve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method

This paper provides an overview of the formulation, analysis and implementation of Spline collocation method for the numerical solution of partial differential equation with two space variable which is of parabolic type. The method includes the solution of non-linear equation which can be expressed as in matrix form. The use of spline collocation methods in the solution of initial-boundary value problems for parabolic-type system id described, with emphasis on alternating direction implicit methods. Problem of vertical groundwater recharge solve by spline collocation method. Finally, recent applications of spline collocation method are outlined

A Six-Step Continuous Multistep Method For The Solution Of General Fourth Order Initial Value Problems Of Ordinary Differential Equations

In this paper, continuous Linear Multistep Method (LMM) for the direct solution of fourth order initial value problems in ordinary differential equation is derived. The study provides the use of both collocation and interpolation techniques to obtain the schemes. Direct form of power series is used as basis function for approximation. An order six symmetric and zero-stable method is obtained. To implement our method, predictors of the same order of accuracy as the main method were developed using Taylor’s series algorithm.  This implementation strategy is found to be efficient and more accurate as the result has shown in the numerical experiments. The result obtained confirmed the superiority of our method over existing schemes Keywords: Direct method; Fourth order; interpolation; collocation multistep methods,;power series; approximate solutions

A 4-Step Implicit Collocation Method for Solution of First and Second Order Odes

ABSTRACT The approach of collocation method approximation will be adopted in the derivation of discrete schemes for direct integration second order ordinary differential equation which are combined together to form a block method. The method is extended to the case in which the approximate solution to a second order (special or general), as well as first order Initial Value Problems(IVPs) can be calculated from the same continuous interpolant and is of order five which is A-stable and has an implicit structure for efficient implementation. The method produces simultaneously approximation of the solution of initial value problems at a block of four points i n x (i=1,2,3,4). Numerical results are given to illustrate the performance method

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain

Fast Isogeometric Boundary Element Method based on Independent Field Approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.Comment: 32 pages, 27 figure

The Hamiltonian BVMs (HBVMs) Homepage

Hamiltonian Boundary Value Methods (in short, HBVMs) is a new class of numerical methods for the efficient numerical solution of canonical Hamiltonian systems. In particular, their main feature is that of exactly preserving, for the numerical solution, the value of the Hamiltonian function, when the latter is a polynomial of arbitrarily high degree. Clearly, this fact implies a practical conservation of any analytical Hamiltonian function. In this notes, we collect the introductory material on HBVMs contained in the HBVMs Homepage, available at http://web.math.unifi.it/users/brugnano/HBVM/index.htmlComment: 49 pages, 16 figures; Chapter 4 modified; minor corrections to Chapter 5; References update

Analytic approximation of solutions of parabolic partial differential equations with variable coefficients

A complete family of solutions for the one-dimensional reaction-diffusion equation $$u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t)$$ with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem for the considered equation with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper the Dirichlet boundary conditions are considered however the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.Comment: 8 pages, 1 figure. Minor updates to the tex