Abstract
This paper considered the formulation of continuous third derivative trigonometrically fitted method for the solution of oscillatory first order initial value problems using the technique of interpolation and collocation of the approximate solution by combining polynomial and trigonometric functions. Solving for the unknown parameters and substituting the results into the approximate solution yielded a continuous linear multistep method, which was evaluated at some selected grid points where two cases were considered at equal intervals to give the discrete schemes which are implemented in block form. The blocks are convergent and stable. Numerical experiments show that the methods compete favorably with existing method.This paper considered the formulation of continuous third derivative trigonometrically fitted block method for the solution of stiff and oscillatory problems. The development of the technique involved the interpolation and collocation of the approximate solution which is the combination of polynomial and trigonometric functions. Solving for the unknown parameters and substituting the results into the approximate solution yielded a continuous linear multistep method, which is evaluated at some selected grid points where two cases were considered at equal intervals to give the discrete schemes which are implemented in block form. The blocks are convergent and stable. Numerical experiments show that the methods compete favorably with existing method and efficient for the solution of stiff and oscillatory problems.
Highlights
Mathematical models are developed to help in the studying 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) [1, 2]
Block method is formulated in terms of linear multistep method (LMM), It preserves the traditional advantage of one step methods, of being selfstarting and permitting easy change of step length [1, 4]
The advantages of block methods over predictor-corrector methods lies in the fact that they are less expensive in terms of number of functions evaluation, it is capable of giving evaluations at different grid points, without overlapping as done in the predictorcorrector method and it generates simultaneous solutions at all grid points [1, 4, 5]
Summary
Mathematical models are developed to help in the studying 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) [1, 2]. We consider numerical procedures for approximating the solution of stiff and oscillatory problems of the form y = f (x, y, ), y (x0) = η0 These type of differential equations are known to be highly oscillatory and some problems have special properties such as discontinuity and stiffness, it is quite difficult to get their numerical solutions accurately [6, 2]. The coefficients of the TFTDM are functions of the frequency and the step-size, the solutions that will be provided by the methods will be highly accurate if (1) has periodic solutions with the unknown frequencies This new method considers the application of higher derivative and does not waste both the computer and human effort which is efficient in handling stiff oscillatory problems. Fitted methods are powerful tool in handling stiff/oscillatory problems, order 2 methods developed by combining polynomial and trigonometric approximate solution and considering higher derivatives, conveniently solved stiff/oscillatory problems and it perform better than other methods of the same order
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