A Numerical Study of Linear Two-Points Boundary Value Problems with ODEs of Fourth-Order Using RKM Method

Mohammed S. Mechee; F. A. Fawzi; Shaymaa Mahmoud Abdullah

doi:10.24237/ASJ.03.01.831C

Authors

Mohammed S. Mechee Information Technology Research and Development Center (ITRDC), University of Kufa, Najaf, Iraq https://orcid.org/0009-0004-8947-5180
F. A. Fawzi Department of Mathematics, Faculty of Computer Science and Mathematics, Tikrit University Salahaddin, Iraq https://orcid.org/0000-0002-3071-8924
Shaymaa Mahmoud Abdullah Department of Mathematics, Faculty of Computer Science and Mathematics, Tikrit University Salahaddin, Iraq https://orcid.org/0000-0002-3071-8924

DOI:

https://doi.org/10.24237/ASJ.03.01.831C

Keywords:

RKM, Ordinary Differential Equations; Boundary Value Problems, Order, DEs, ODEs.

Abstract

This study established the numerical RKM approach for solving linear two-point boundary
value problems involving fourth-order ordinary differential equations. However, the proposed
developed numerical RKM method has been tested using some implementations in order to
compare it with the exact solutions to establish the method's validity. Furthermore, this
comparison demonstrates that the proposed direct integrator is more efficient than the indirect
method in terms of efficiency and accuracy. In addition, numerical implementations are used
in order to show the efficiency and time-based complexity of function evaluations. This direct
method's suggested strategy, which has wonderful qualities like quick and efficient calculation,
also requires fewer computational employees.

References

N. T. Yen, Computational methods in engineering boundary value problems, Academic

press, (1980)

G. Wojciech, The use of Walsh-wavelet packets in linear boundary value problems,

Computers & structures, Elsevier, 82(2-3), 131-141(2004),

DOI(https://doi.org/10.1016/j.compstruc.2003.10.004)

N. M. Aslam, T. Ikram, K. M. Azam, Quadratic non-polynomial spline approach to the

solution of a system of second-order boundary-value problems, Applied Mathematics

and Computation, Elsevier, 179(1), 153-160(2006),

DOI(https://doi.org/10.1016/j.amc.2005.11.091)

T. Ikram, K. M. Azam, Quartic non-polynomial spline approach to the solution of a

system of third-order boundary-value problems, Journal of mathematical analysis and

applications, Elsevier, 335(2), 1095-1104(2007),

DOI(https://doi.org/10.1016/j.jmaa.2007.02.046)

T. Ikram, Nonpolynomial spline approach to the solution of a system of second-order

boundary-value problems, Applied Mathematics and Computation, Elsevier, 173(2),

-1218(2006), DOI(https://doi.org/10.1016/j.amc.2005.04.064)

S. Robert, J. Shipman, Solution of Troesch’s two-point boundary value problems by

shooting techniques, Journal of Computational Physics, 10, 232-241(1972),

DOI(https://doi.org/10.1016/0021-9991(72)90063-0)

M. Tatari, M. Dehghan, The use of the Adomian decomposition method for solving

multipoint boundary value problems, Physica Scripta, IOP Publishing, 73(6), 672(2006),

DOI(10.1088/0031-8949/73/6/023)

T. Ikram, E. H. Twizell, Higher-order finite-difference methods for nonlinear secondorder two-point boundary-value problems,Applied Mathematics Letters, Elsevier, 15,

(7), 897-902(2002), DOI(https://doi.org/10.1016/S0893-9659(02)00060-5)

T. Ikram, K. M. Azam, Non-polynomial splines approach to the solution of sixth-order

boundary-value problems, Applied mathematics and computation, Elsevier, 195(1),

-284(2008), DOI(https://doi.org/10.1016/j.amc.2007.04.093)

M. Mechee, N. Senu, F. Ismail, B. Nikouravan, Z. Siri, A three-stage fifth-order RungeKutta method for directly solving special third-order differential equation with

application to thin film flow problem Mathematical Problems in Engineering, (2013),

DOI(https://doi.org/10.1155/2013/795397)

F. A. Fawzi, N. Senu, F. Ismail, Z. A. Majid, A new integrator of Runge-Kutta type for

directly solving general third-order odes with application to thin film flow

problem, Appl. Math, 12(4), 775-784(2018),

DOI(http://dx.doi.org/10.18576/amis/120412)

F. A. Fawzi, N. Senu, F. Ismail, Z. A. Majid, An efficient of direct integrator of RungeKutta type method for solving y‴= f (x, y, y′) with application to thin film flow

problem, International Journal of Pure and Applied Mathematics, 120(1), 27-50(2018),

DOI(10.12732/ijpam.v120i1.3)

M. S. Mechee, M. A. Kadhim, Direct explicit integrators of RK type for solving special

fourth-order ordinary differential equations with an application, Global Journal of Pure

and Applied Mathematics, 12(6), 4687-4715(2016)

M. S. Mechee, M. A. Kadhim, Explicit direct integrators of R type for solving special

fifth-order ordinary differential equations, American Journal of Applied

Sciences, 13(12), 1452-1460(2016),

DOI(https://doi.org/10.3844/ajassp.2016.1452.1460)

M. S. Mechee, F. A. Fawzi, Generalized Runge-Kutta integrators for solving fifth-order

ordinary differential equations, Italian J. Pure Appl. Math, 45, 600-610(2021)

M. S. Mechee, Generalized RK integrators for solving class of sixth-order ordinary

differential equations, Journal of Interdisciplinary Mathematics, 22(8), 1457-

(2015), DOI(https://doi.org/10.1080/09720502.2019.1705502)

M. S. Mechee, J. K. Mshachal, Derivation of embedded explicit RK type methods for

directly solving class of seventh-order ordinary differential equations, Journal of

Interdisciplinary Mathematics, Publisher of Taylor and Francis, 22(8), 1451-

(2019), DOI(https://doi.org/10.1080/09720502.2019.1700936)

M. S. Mechee, K. Ben. Mussa, Generalization of RKM integrators for solving a class of

eighth-order ordinary differential equations with applications, Advanced Mathematical

Models and Applications, 5(1), 111-120(2020)

M. S. Mechee, H. M. Wali, K. Ben. Mussa, Developed RKM Method for Solving NinthOrder Ordinary Differential Equations with Applications, Journal of Physics:

Conference Series, IOP Publishing, 1664(1), 012102(2021), DOI(10.1088/1742-

/1664/1/012102)

M. S. Mechee, J. K. Mshachal, F. A. Fawzi, Derivation of direct explicit methods of

RKM-type for solving special class of tenth-order ordinary differential equations, In:

AIP Conference Proceedings, AIP Publishing , 2414(1), (2023),

DOI(https://doi.org/10.1063/5.0114820)

N. Senu, M. Mechee, F. Ismail, Z. Siri, Embedded explicit Runge–Kutta type methods

for directly solving special third order differential equations y‴= f (x, y), Applied

Mathematics and Computation, 240, 281-293(2014),

DOI(https://doi.org/10.1016/j.amc.2014.04.094)

M. Y. Turki, F. Ismail, Z. B. Ibrahim, N. Senu, Two and Three point Implicit Second

Derivative Block Methods For Solving First Order Ordinary Differential Equations,

(2018).

M. Y. Turki, F. Ismail, N. Senu, Z. B. Ibrahim, Second derivative multistep method for

solving first-order ordinary differential equations, In: AIP Conference Proceedings,

(1), AIP Publishing LLC, 2020(2016),

DOI(https://ui.adsabs.harvard.edu/link_gateway/2016AIPC.1739b0054T/doi:10.1063/

4952534)

M. Y. Turki, M. M. Salih, M. S. Mechee, Construction of General Implicit-Block

Method with Three-Points for Solving Seventh-Order Ordinary Differential Equations,

Symmetry, MDPI publisher, 14(8), 1605(2022),

DOI(https://doi.org/10.3390/sym14081605)

D. Faires, Burden, Numerical Methods, (Inc., Pacific Grove, publisher Thomson

Learning, 2003)

A Numerical Study of Linear Two-Points Boundary Value Problems with ODEs of Fourth-Order Using RKM Method

Authors

DOI:

Keywords:

Abstract

References

Downloads

Published

How to Cite

Issue

Section

License

Befor Submitting

Make a Submission

Academic Scientific Journals

DOAJ

info