A New Hybrid Block Method via Combined Hermite Polynomials and Exponential Functions as Basis Function

Year 2023, Volume: 5 Issue: 2, 10 - 23, 20.11.2023

Hycienth Orapine , Zirra Donald Ali A. Baidu Joshua Oladele

https://doi.org/10.54286/ikjm.1227629

Abstract

This paper presents the derivation and implementation of a hybrid block method for solving stiff and oscillatory first-order initial value problems of ordinary differential equations (ODEs). The hybrid block method was derived by continuous collocation and interpolation using combined Hermite polynomials and exponential functions as the basis function to produce a continuous implicit Linear Multistep Method (LMM) of order nine and implement in block form. The basic properties of the derived method were studied, and the hybrid block integrator was demonstrated to be zero-stable, convergent, consistent, and to have an A-stable region of absolute stability, which made it suitable for stiff and oscillatory ordinary differential equations. The use of a combined basis in the generation of LMMs is worthy of universal acceptance. The technique indicates that, utilizing an interpolation and collocation approach, continuous LMMs can be derived from combinations of any polynomials and exponential functions. On two sampled stiff and oscillatory problems, the new integrator was tested. The numerical results indicate that our new hybrid block integrator is computationally efficient and outperforms existing methods in terms of accuracy

Keywords

Basis function, collocation, exponential function, hermite Polynomials, hybrid block method, interpolation, Ordinary Differential Equations

References

• Aboiyar, T., Luga, T., & Iyorter B.V. (2015). Derivation of Continuous Linear Multistep Methods Using Hermite Polynomials as Basis Functions. American Journal of Applied Mathematics and Statistics, 3(6):220-225, doi:10.12691/ajams-3-6-2.
• Adesanya, A.O., Sunday, J. & Momoh, A.A. (2014). A New Numerical Integrator for the Solution of General Second Order Ordinary Differential Equations. International Journal of Pure and Applied Mathematics, 97(4):431-445.
• Butcher, J. C. (2003). Numerical Methods for Ordinary Differential Equation West Sussex: John Wiley & Sons.
• Campbell S.L. and Heberman R. (2011). Introduction to Differential Equations with Dynamical Systems. New Jersey: Princeton University Press. Chap. 1, pp. 1-2.
• Dahlquist, G. G. (1956). Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4:33-50.
• Fatunla SO. (1988). Numerical methods for initial value problems in ordinary differential Equations. New York: Academic Press Inc;
• Fotta A. U., Bello A. and Shelleng Y.I. (2015). Hybrid Block Method for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. IOSR Journal of Mathematics, 2(6):60-66.
• Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations. New York: John Wiley and Sons.
• Momoh, A. A., Adesanya, A. O., Fasasi, K. M., and Tahir, A. (2014). A New Numerical Integrator for the Solution of Stiff First Order Ordinary Differential Equations. Engineering Mathematics Letters 5: http://scik.org.
• Odekunle, M. R., Adesanya A. O. and Sunday, J. (2012). 4-Point block method for direct integration of first-order ordinary differential equations, International Journal of Engineering Research and Applications, 2:1182-1187.
• Okunuga, S.A. & Ehigie, J. (2009). A New Derivation of Continuous Collocation Multistep Methods Using Power Series as Basis Function. Journal of Modern Mathematics and Statistics, 3(2):43-50.
• Solomon, G and Hailu, M. (2020). A Seven-Step Block Multistep Method for the Solution of First Order Stiff Differential Equations. Momona Ethiopian Journal of Science 12(1):72-82.
• Sunday, J., Odekunle, M.R., James A.A. & Adesanya, A.O. (2014). Numerical Solution of Stiff and Oscillatory Differential Equations Using a Block Integrator. British Journal of Mathematics & Computer Science. 4(17): 2471-2481.
• Yan YL. (2011). Numerical methods for differential equations. Kowloon: City University of Hong Kong.