A 3-Step Single Algorithm Developed for Direct solution of Linear and Nonlinear First, Second, Third, Fourth and Fifth Orders of Ordinary Differential Equations

Musa G. Garba, Simon Daniel, Peter Ayuba, Dauda M. K.

Abstract


A single algorithm has been developed to efficiently solve ordinary differential equations (ODEs) ranging from first to fifth order. This method utilizes interpolation and collocation techniques, with a power series function as its foundation. The algorithm accurately places derivatives at grid and off-grid points, except for the end points, and has undergone thorough examination of its fundamental properties. Results indicate that this novel approach exhibits superior accuracy when compared to existing methods for solving ODEs of these orders. Moreover, it eliminates the need for developing separate methods for each of the specified five categories of ODEs.


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References


Abolarin et al (2020). Derivation and Implementation of a three –step Hybrid Block Algorithm (TSHBA) for direction solution of Linear and Nonlinear Second order ODEs. FUDMA Journal of Sciences (FJS).4(4):477-483.

Adeyefa, E. O., & Kuboye J. O. (2020). Derivation of New Numerical Model capable of solving Second and Third Order Ordinary Differential Equations. IAENG International Journal of Applied Mathematics. 50:2

Dauda et al (2022). Second Derivative Block Hybrid Methods for the Numerical Integration of Differential systems. Journal of Fractal and Fractional. 6:1-16

Ezekiel, O. O. & Bamikole, G. O. (2018). 3-Point Single Hybrid Block Method (3PSHBM) for the direct solution of General Second order initial value problem of ODEs. Journal of Scientific Research & Reports, 20(3):1-11

Faruk et al (2023). Coherent Hybrid Block Method for Approximating Fourth order ODEs. Journal of Amasya Institute of Sciences and Technology. 4(1):1-19

Fatunla, S. O. (1988). Numerical methods for initial value problem in ordinary Differential Equations. Academic Press inc. Harcourt Brace Jovanovich Publishers, New York

Fatunla, S. O. (1991). Block methods for second order ODEs. International Journal of Computer Mathematics, 41,55-63

Fatunla, S. O. (1994). A Class of Block Method for second order initial value Problems. International Journal of Computational Mathematics, 55, 119-133

Friday, O. O. (2023). Three –step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations. Asian Research Journal of Mathematics 19(6), 25-44.

Jain, M. K. (1978), Numerical solution of differential equations 2nd edn.

John et al (2022). Single Numerical Algorithm developed to solving First and Second ODEs. International Journal of Mathematics in Operational Research. 21(4):466-479

Joshua et al (2021). A pair of three-step Hybrid Block methods for the solutions of Linear and Nonlinear First-order systems. Utilitas Mathematica 118:1-15

Joshua et al (2022). An Accurate Preserving Block Hybrid Algorithm for the Integration of Second-order Physical Systems with Oscillatory Solutions. Journal of the Nigerian Society of Physical Sciences

Kuboye, J.O. & Omar, Z. (2015). Multistep Collocation Block Method for Direction Solution of Second Order Ordinary Differential Equations. American Journal of Applied Sciences, 12(9): 663-668

Lambert, J. D. (1973). Computational methods in Ordinary Differential Equations, John Wiley & Sons London.

Lambert, J. D. (1991). Numerical Methods for in Ordinary Differential Systems, the initial value Problem. John Wiley and sons, New York

Morteza, B. & Higinio, R. (2023). A computational Block Method with Five Hybrid points for Differential Equations containing a piece-wise constant Delay. Journal of Differential Equations and Dynamical systems

Muhammad, et al (2014). Derivation and analysis of block implicit hybrid backward differentiation formulae for stiff problems. Nigerian Journal of Mathematics and Applications vol. 23

Muideen et al (2021). Two-Step Bi-basis Hybrid Block Method for direct Approximation of Fourth Order ODEs with intermediate Bi-intra Points. Journal of Applied Sciences, information and computing. 2(1):1-10.

Muideen et al (2023). Generalization of a class of uniformly optimized k-step hybrid block method for solving two-point boundary value problems. Results in Physics 44:106147

Ogunniran et al (2022). An Accurate Hybrid Block Technique for Second Order Singular problems in Ordinary Differential Equations. African Journal of Pure & Applied Sciences. 144-154.

Olusola, E. A. & Bamikole, G. O. (2022). New hybrid method for director numerical solution of nonlinear Second, third and fourth orders ordinary differential equations.

Oluwaseun, A. & Zurni, O. (2019). Solving Third Order Ordinary Differential Equations using One-Step Block Method with four Equidistant Generalized Hybrid Points. International Journal of Applied Mathematics

Omar, Z. & Kuboye, J.O. (2015). Derivation of Block Methods for solving Second-Order Ordinary Differential Equations Directly Using Direct Integration and Collocation Approaches. Indian Journal of Science and Technology, 8(12)

Roseline, et al (2021). World Journal of Advance Research and Review. 09(01), 239-249.

Raymond, D. & Kygya, Y. T. (2020). Three-Step Two-Hybrid Block Method for the Direct Solution of Second- Order Ordinary Differential Equations.

Victor, O. A. & Solomon, O. A. (2021). A New Special 15-Step Block Method for solving for solving General Fourth order ODEs. Journal of the Nigerian Society of Physical Sciences. 3:308-333.


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