A COMPUTATIONAL ERROR ESTIMATION OF THE INTEGRAL FORMULATION OF THE TAU METHOD FOR SOME CLASS OF ORDINARY DIFFERENTIAL EQUATIONS

Well Researched and Ready to use Ph.D Thesis, page numbers: 227, Department: Mathematics

ABSTRACT 

The tau method for the numerical solution of Ordinary Differential Equations (ODEs) seeks an approximant of the solution which exactly satisfies the corresponding Perturbed Ordinary Differential Equations (PODE). This tau approximant has the power of x as its basis function. The integrated formulation of the tau method was developed to improve the accuracy of the approximant. It is therefore desirable to generalize this variant as a forerunner to the automation of the technique. This desire led us to obtain some general results for the variant and its error estimate for the class of ODEs with maximum of six tau parameters in the perturbation terms. The error estimation was based on the error of Lanczos economization process and it satisfies the corresponding Perturbed Error Differential Equation (PEDE). We integrated through this PEDE and consequently increased the order of the perturbation term leading to an increased accuracy of the result obtained. Members of the class of problems characterized in (ii) were investigated for study. Consequently, some results which generalize tau method and its error estimate for the integrated variant were obtained. This was implemented on some problems to test the accuracy of the estimates. The results were that: (i) a generalized tau matrix system was constructed for m-th order linear ODEs; (ii) general error estimates for the class of problems with maximum of six overdetermination were obtained; (iii) development of a tau program for the integrated formulation of tau method; and (iv) extension of the work to nonlinear problems; (v) the error estimate accurately captures the order of the tau approximant. This study shows an improvement over some earlier works on the error estimation of the tau methods as the higher order perturbation term in the integrated variant has brought about better accuracy of the resulting variant and error estimate without involving matrix inversion, thus making it attractive for an efficient numerical algorithm in the solution of ordinary differential equations. The error estimation of the tau method for three variants of the method is discussed, using some appropriate examples for the purpose of comparing the accuracy of the estimates obtained for each of them. However, it has been noted that the integrated formulation gave the least error due to the higher order perturbation term (Hn+m(x)) it involved and both differential and recursive forms yielded approximation of the same order. We also consider the solution of the initial value problems for first order ODEs using the tau methods.

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