### Four step hybrid block method for the direct solution of fourth order ordinary differential equations

Document Type : Research Paper

Author

Department of Mathematics‎, College of Art and Sciences-Tabarjal‎, Jouf University, Saudi Arabia

Abstract

‎This paper proposes a direct four-step implicit hybrid block method for directly solving general fourth-order initial value problems of ordinary differential equations‎. ‎In deriving this method‎, ‎the approximate solution in the form of power series is interpolated at four points‎, ‎i.e $x_n, ‎x_{n+1},x_{n+2},x_{n+3}$ while its forth derivative is collocated at all grid points‎, ‎i.e $x_n‎,x_{n+\frac{1}{4}},‎x_{n+1}‎ , x_{n+2}‎, x_{n+\frac{5}{2}}‎, x_{n+3}‎,x_{n+\frac{7}{2}}$ and $x_{n+4}$ to produce the main continuous schemes‎. ‎In order to verify the applicability of the new method‎, ‎the properties of the new method such as local truncation error‎, ‎zero stability‎, ‎order and convergence are also established‎. ‎The performance of the newly developed method is then compared with the existing methods in terms of error by solving the same test problems‎. ‎The numerical results reveal that the proposed method produces better accuracy than several existing methods when solving the same initial value problems (IVPs) of second order ODEs‎.

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