Diagonally implicit multistep block method for solving delay volterra integro-differential equation
In this study, two points diagonally implicit multistep block (2DIMB) methods are constructed for the numerical solution of the first and second order delay Volterra integro-differential equation (DVIDE). The second order of DVIDE is solved directly without reducing the problem in the system of t...
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| Format: | Thesis |
| Language: | English English |
| Published: |
2023
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/111577/1/IPM%202023%203%20-%20IR.pdf http://psasir.upm.edu.my/id/eprint/111577/ |
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| Summary: | In this study, two points diagonally implicit multistep block (2DIMB) methods
are constructed for the numerical solution of the first and second order delay
Volterra integro-differential equation (DVIDE). The second order of DVIDE is
solved directly without reducing the problem in the system of the first order of
DVIDE. Two distinct types of DVIDE are solved, namely unbounded and bounded
time lag cases. Furthermore, the constant and pantograph delay types indicate that
the delay conditions for DVIDE are also considered in this study. The strategy
of the constant step size is implemented for finding the numerical solution to DVIDE.
When finding the approximate solution to DVIDE, three components must be
considered: the initial value problem of DVIDE, the delay solution, and the integral
part. The 2DIMB method is formulated for the numerical solution of initial value
problem of DVIDE and computed two solutions simultaneously in block form. This
method is built on a predictor-corrector formula.
The previously calculated solutions are used to obtain the delay solution for the constant
delay type. Meanwhile, Lagrange interpolation polynomial is implemented to
approximate the delay solution for the pantograph delay type. Since an integral part
of DVIDE cannot be solved explicitly and analytically, the idea of approximating
the solution is discussed. The appropriate order of the numerical integration method
is chosen to approximate the solution of the integral part of DVIDE, which include
trapezoidal rule, Simpson’s rule, and Boole’s rule.
Analysis on order, error constants, consistency, zero-stability, and convergence
of the proposed method are given in this study. Moreover, the stability region is
discussed based on the stability polynomial of the 2DIMB method paired with the
appropriate numerical integration method. All the computational procedures were
undertaken using the C programming language in a CODE::BLOCKS platform.
Numerical results showed that where the proposed methods are reliable and suitable
for solving the unbounded and bounded time lag of the DVIDE for the constant and
pantograph delay types. Three advantages in terms of the total steps taken, function
evaluations and the execution time taken by these methods have been identified
when comparing the numerical results with the Runge-Kutta and Adam-Bashforth-
Moulton methods. |
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