Journal of Approximation Theory and Applied Mathematics - 2015 Vol. 5

Marco Schuchmann

• Verlag: Books on Demand
• Jahr: 2016
• ISBN drucken:
  9783739246628
• eBook-ISBN:
  9783741255304
• Auflage: 3
• Thema: Mathematik
• Thema: Mathematik Allgemein

Journal of Approximation Theory and Applied Mathematics (ISSN 2196-1581) is a journal which started in 2013.Themes of our journal are: Approximation theory (with a focus on wavelets) and applications in mathematics like numerical analysis, statistics or financial mathematics.

Testing an Algorithm for an Adaptive Wavelet Based Method in Mathematica and Comparing the Method with NDSolve

M. Schuchmann and M. Rasguljajew from the Darmstadt University of Applied Sciences

Abstract

M. Schuchmann developed in 2013 and 2014 an adaptive wavelet based algorithm to numerically solve differential equations. This algorithm can be applied to ODEs with various orders and even to PDEs and differential-algebraic equations (DAEs). In this Paper we compare the solutions of boundary value problems of second order ODEs calculated by the adaptive wavelet algorithm with the solution of the Mathematica function NDSolve. In NDSolve we have different explicit or implicit integration methods (like implicit or explicit Runge-Kutta, Adams or BDF method) and it will automatically reduce the step size or change the method depending on the problem, so NDSolve can handle stiff and non-stiff problems.

We have implemented the algorithm in a Mathematica module and applied several tests. The used wavelet collocation method is costly, like collocation method are costly in general. But for stiff or unstable problems it has an advantage and for these cases explicit methods cannot be used. We used the Shannon wavelet in our module, but it can be modified for other wavelets and the ODE can be given in implicit form.

Introduction

Our module can be used for ODEs of order 1 and 2. It can easily be adjusted for higher order ODEs. It uses an adaptive algorithm to adjust the parameters of the wavelet based method.

The method uses an approximation function y_j constructed with a subset of the bases of V_j. With that approximation function we minimize the sum of squares of the residuals at so called collocation points t_i. For a certain number of collocation points this method is equivalent to the collocation method using that ‘wavelet bases’. The initial points or boundary values are integrated in the minimization process, but they could also be used as restraints.

Here we have to adjust several parameters like the solution index j, the number of bases functions out of V_j and the number and the position of the collocation points t_i. The adjustment of these parameters is the objective of the adaptive algorithm. For the construction of the algorithm theoretical investigations have been necessary and M. Schuchmann has used an error estimation (see [16]) which is based on the criteria Q_a (see (2)) and he also evaluated many simulation to use the information of the minimum of Q (see (1)...

Inhaltsverzeichnis

• Inhaltsverzeichnis
• New Methods of Approximation of Step Functions
• Resolution of Nonlinear Partial Differential Equations by Elzaki Transform Decomposition Method
• Testing an Algorithm for an Adaptive Wavelet Based Method in Mathematica and Comparing the Method with NDSolve
• Appendix
• Impressum