EXPONENTIALLY-FITTED NUMERICAL INTEGRATION SCHEMES FOR FIRST AND SECOND ORDER DIFFERENTIAL EQUATIONS
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Date
2021-04
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UNIVERSITY OF ILORIN
Abstract
Mathematical models are developed to help in the understanding of physical phenomena. These models often yield equations that contain some derivatives of an unknown function of one or several variables. Such equations are called Differential Equations (DEs). A DE in which the unknown function is a function of two or more independent variables is called a Partial Differential Equation (PDE). Those in which the unknown function is a function of only one independent variable are called Ordinary Differential Equations (ODEs). This work concerns the study of numerical solutions of both types. Since most DEs do not have closed form solution, the development of numerical techniques becomes necessary. Methods used for solving Initial Value Problems ( IVPs ) of general ODEs have been reported to have setbacks as regards to handing certain attributes like singularity of stiffness and oscillatory nature of problems. Furthermore, implementation via predictor-corrector mode in Linear Multistep Methods (LMMs) have been found to be exhausting in terms of the number of function evaluations per step. K-step exponentially-fitted block methods circumvent these challenges associated with
LMMs, for solving IVPs, Boundary Value Problems (BVPs) and Initial Boundary Value Problems (IBVPs) of special nature; hence the motivation of this work.
This study is therefore aimed at developing a class of efficient numerical integration schemes, for direct solution of first order IVPs, second order BVPs and IBVPs. To this end, the objectives were to: (i) construct trial solution with power series fitted with exponential function as basis function; (ii) develop new classes of k-step continuous
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exponentially-fitted block methods; (iii) develop continuous 2-step hybrid exponentiallyfitted block method; (iv) analyze the methods for zero stability, consistency and convergence; and (v) compare the accuracy of the proposed schemes with those of existing methods using some existing problems. The first order IVP of the form
y0 = f(x,y), a < x < b (1)
and the second order BVPs of the form
y00 = f(x,y,y0), a < x < b (2)
with appropriate boundary conditions were considered. The trial solution of the form
k
p(x) = Xaixi + ak+1ewx
i=0 (3)
∼= y(x)
was assumed where the ai’s are unknown constants and ω is the frequency. In order to determine the unknown constants, interpolation and collocation methods were employed. The resulting values were later substituted back into (3), which after some algebraic manipulations yielded the desired continuous exponentially fitted block method.
The findings of this study were;
(i) new classes of k-step exponentially fitted block methods for direct solution of first order IVPs and second order BVPs were derived;
(ii) new 2-step Continuous hybrid exponentially fitted block method for solving second order IBVPs were developed;
(iii) the methods are zero stable, consistent and convergent;
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(iv) the methods have small global error and less computation time compared to existing methods; and
(v) the methods are self-starting and admit easy change of step-sizes and function evaluation at off grid points.
The study concluded that an efficient class of continuous and discrete numerical integration schemes of implicit form for first and second order problems were developed and successfully implemented. These schemes are therefore recommended for solution of first and second order IVPs, second order BVPs and IBVPs.
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Keywords
EXPONENTIALLY-FITTED, NUMERICAL INTEGRATION SCHEMES, FIRST ORDER DIFFERENTIAL EQUATIONS, SECOND ORDER DIFFERENTIAL EQUATIONS