Sixth-Order Hybrid Boundary Value Method for Systems of Boundary Value Problems

PDF

ABSTRACT

Hybrid Boundary Value Methods (HyBVMs) are a new class of Boundary Value Methods (BVMs) proposed recently for the approximation of Ordinary Differential Equations (ODEs). Just like the BVMs, the HyBVMs are also based on the Linear Multistep Methods (LMMs) while utilizing data at both step and off-step points. Numerical tests on both linear and nonlinear Boundary Value Problems (BVPs) were presented using HyBVMs of order 6. The results were compared with two symmetric schemes: Extended Trapezoidal Rule (ETR) and Top Order Method (TOM).

KEYWORDS

Boundary value methods, hybrid BVMs, boundary value problems, linear multistep method, numerical methods for ODEs.

REFERENCES

[1] S. N. Jator and J. Li, “Boundary Value Methods via a Multistep Method with Variable Coefficients for Second Order Initial and Boundary Value Problems”, International Journal of Pure and Applied Mathematics, 50, (2009), 403-420.

[2] F. Mazzia, “Boundary Value Methods for the Numerical Solution of Boundary Value Problems in Differential-Algebraic Equations”. Bolletino della Unione Matematica Italiana, (1997), 579-593.

[3] L. Brugnano and D. Trigiante, “Block Boundary Value Methods for Linear Hamiltonian Systems”. Appl. Math. Comput., 81, (1997), 49-68.

[4] P. Amodio and F. Iavernaro, “Symmetric boundary value methods for second initial and boundary value problems”. Medit. J. Maths., 3, (2006), 383-398.

[5] T. A. Biala and S. N. Jator, “A boundary value approach for solving three-dimensional elliptic and hyperbolic Partial Differential Equations”, SpringerPlus Journals, vol. 4, article no 588, 2015

[6] P. Amodio and F. Mazzia, “A Boundary Value Approach to the Numerical Solution of Initial Value Problems by Multistep Methods”, J Difference Eq. Appl., 1, (1995), 353-367.

[7] L. Aceto, P. Ghelardoni and C. Magherini, “PGSCM: A family of PStable Boundary Value Methods for Second Order Initial Value Problems”, Journal of Computational and Applied Mathematics, 236, (2012), 3857-3868.

[8] T. A. Biala, S. N. Jator and R. B. Adeniyi, “Numerical approximations of second order PDEs by boundary value methods and the method of lines”, Afrika Matematika, 2016.

[9] L. Brugnano and D. Trigiante, “Convergence and Stability of Boundary Value Methods for Ordinary Differential Equations”, Journal of Computational and Applied Mathematics, 66, (1996), 97-109.

[10] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1956.

[11] T. A. Anake, Continuous implicit hybrid one-step methods for the solution of initial value problems of general second-order ordinary differential equations, Ph.D Thesis. Covenant University, Nigeria.

[12] J. C. Butcher, “A modified multistep method for the numerical integration of ordinary differential equations”, J. Assoc. Comput. Mach., 12, (1965), 124-135.

[13] A. K. Ezzeddine and G. Hojjati, “Hybrid extended backward differentiation formulas for stiff systems”, Internation Journal of Nonlinear Science, 1292, (2011), 196-204.

[14] T. A. Anake, D. O. Awoyemi and A. A. Adesanya, “A one step method for the solution of general second order ordinary differential equations”, International Journal of Science and Technology, 2(4), (2012), 159-163.

[15] C. W. Gear, “Hybrid Methods for Initial Value Problems in Ordinary Differential Equations”, Math. Comp., 21, (1967), 146-156.

[16] W. B. Gragg and H. I. Stetter, “Generalized multistep predictorcorrector methods”, J. Assoc. Comput Mach., 11, (1964), 188-209.

[17] L. Brugnano and D. Trigiante, “High-Order Multistep Methods for Boundary Value Problems”, Applied Numerical Mathematics, 18, (1995), 79-94.

[18] L. Brugnano and D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science, Amsterdam, 1998.

[19] R. S. Stepleman, “Triadiagonal Fourth Order Approximations to General Two-Point Nonlinear Boundary Value Problems with Mixed Boundary Conditions”, Mathematics of Computation, 30, (1976), 92- 103.

Cite this paper

Grace O. Akinlabi, Raphael B. Adeniyi. (2018) Sixth-Order Hybrid Boundary Value Method for Systems of Boundary Value Problems. International Journal of Mathematical and Computational Methods, 3, 16-19