Two-point diagonally implicit multistep block method for solving volterra integro-differential equation of second kind
The first part of the thesis focuses on solving Volterra integro-differential equation (VIDE) of the second kind with the multistep block method. The two points diagonally implicit multistep block (2PDIB) method is formulated for the numerical solution of the second kind of VIDE. The derivatio...
محفوظ في:
| المؤلف الرئيسي: | |
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| التنسيق: | أطروحة |
| اللغة: | English |
| منشور في: |
2018
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| الموضوعات: | |
| الوصول للمادة أونلاين: | http://psasir.upm.edu.my/id/eprint/69426/1/IPM%202018%205%20IR.pdf |
| الوسوم: |
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| الملخص: | The first part of the thesis focuses on solving Volterra integro-differential equation
(VIDE) of the second kind with the multistep block method. The two points
diagonally implicit multistep block (2PDIB) method is formulated for the numerical
solution of the second kind of VIDE. The derivation of the 2PDIB method can
be obtained using Lagrange interpolating polynomial. The numerical solution of
the second kind of VIDE computed at two points simultaneously in block form
using the proposed method using constant step size. These numerical solutions are
executed in the predictor-corrector mode.
Since an integral part of VIDE cannot be solved explicitly and analytically,
the idea to approximate the solution of the integral part is discussed and the
appropriate order of numerical integration formulae is chosen to approximate the
solution of the integral part of VIDE which include trapezoidal rule, Simpson’s
rule and Boole’s rule. Regarding the general form of VIDE, there are two cases
of the kernel which are K(x; s) = 1 and K(x; s) ≠= 1. Two different procedures are
developed to obtain the solution for these cases. The stability region is discussed
based on the stability polynomial of the 2PDIB method paired with the appropriate
quadrature rule.
Linear and nonlinear problems of VIDE have been solved numerically using
the 2PDIB method. Six tested problems are presented in order to study the performance and efficiency of the 2PDIB method in terms of maximum error, total
function calls, total steps taken and the execution time taken. Numerical results
showed that the efficiency of 2PDIB method when solving VIDE compared to the
existing methods. |
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