Abstract
A new two-step modified explicit hybrid method with parameters depending on the step-size is constructed. This method is derived using the coefficients from a sixth-order explicit hybrid method with extended interval of absolute stability and then imposed each stage of the modified formula to exactly integrate the differential equations with solutions that can be expressed as linear combinations of sinwx and coswx, where w is the known frequency. Numerical results show the advantage of the new method for solving oscillatory problems.
Highlights
Second-order ordinary differential equations are important tools for modelling physical phenomena in science and engineering. is paper is concerned with the numerical solution of the second-order ordinary differential equations of the form y′′(x) f(x, y(x)), y x0 y0, y′ x0 y0′, (1)
We develop a sixth-order explicit hybrid method with extended interval of absolute stability based on the class of explicit hybrid methods (2)
Consider the modified four-stage explicit hybrid method represented by this tableau:
Summary
Second-order ordinary differential equations are important tools for modelling physical phenomena in science and engineering. is paper is concerned with the numerical solution of the second-order ordinary differential equations of the form y′′(x) f(x, y(x)), y x0 y0, y′ x0 y0′, (1). Is paper is concerned with the numerical solution of the second-order ordinary differential equations of the form y′′(x) f(x, y(x)), y x0 y0, y′ x0 y0′, (1). In the case of adapted numerical methods, particular algorithms have been proposed by several authors, see [1,2,3] to solve these classes of problems. Franco [4] has established the following class of explicit hybrid methods: Y1 yn− 1, Y2 yn, i− 1. We develop a sixth-order explicit hybrid method with extended interval of absolute stability based on the class of explicit hybrid methods (2). Using the coefficients from the sixth-order hybrid method, we develop a new modified hybrid method.
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