Abstract
A Block Second Derivative Formula (BSDF) of order seven for the numerical solution of Hessenberg Differential Algebraic Equations (DAEs) of index 3 is presented. This is achieved by constructing a Continuous Second Derivative Formula (CSDF) used to obtain the main and additional methods which are combined to form a single block of methods that simultaneously provide the approximate solutions for the Hessenberg DAEs of index 3. The error constant and stability properties of the BSDF are discussed. The performance of the method is demonstrated on some numerical examples to show the accuracy and efficiency of the method. AMS Subject Classification: 65L05, 65L06, 65L08
Highlights
Many important mathematical problems can be modeled as systems of Differential-Algebraic Equations (DAEs)
The aim in this paper is to develop a Block Second Derivative formula (BSDF) of order k+2 for the solution of Hessenberg DAE of the form (5) that will be efficient, reliable and accurate
We construct the continuous scheme of the main and additional methods known as the second derivative formula (SDF) and are combined to form the Block Second Derivative Formula (BSDF) on the interval from tn to tn+k = tn + kh where h is the chosen step-length and k is the step number of the method
Summary
Many important mathematical problems can be modeled as systems of Differential-Algebraic Equations (DAEs) These problems have a wide range of applications in various branches of science and engineering. Okunuga some numerical methods have been developed for the solution of DAEs such as the BDF (see [2], [3], [9]), implicit Runge-Kutta methods [3], Pade and Chebysev approximation methods (see [4] [5] [6]) and variational iterative method [12] These methods are only directly suitable for low-index problems and often require that the problem, have special structure. Akinfenwa and Okunuga [1] and Naghmeh et al.[15] developed block method for the solution of semi explicit index 1 DAEs This paper as in [1] presents a block method which preserves the Runge-Kutta traditional advantage of being self-starting and efficient, since it requires m function evaluations per integration step, where m is the number of functions in the block method
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