Direct multistep method for solving retarded and neutral delay differential equation with boundary and initial value problems. (English) Zbl 1541.65051
Summary: The boundary and initial conditions that are related to the retarded and neutral delay differential equations, respectively, will be resolved in this work by using the previous direct multistep method. This method solves retarded and neutral delay differential equations directly by implementing the proposed method without converting it to a first-order system. For boundary value problems, the shooting strategy incorporated with the Newton method is utilized to predict the guessing value. The initial value problem for neutral delay differential equations on the other hand is resolved directly with special attention to the differential part of the problem. Several numerical examples are investigated to observe the capability of the developed strategies and methods for solving retarded delay differential equations with boundary value problems and neutral delay differential equations with initial value problems.
MSC:
| 65L03 | Numerical methods for functional-differential equations |
| 34K40 | Neutral functional-differential equations |
| 34K10 | Boundary value problems for functional-differential equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
Keywords:
boundary value problem; constant delay; retarded delay differential equation; direct method; initial value problem; neutral delay differential equation; shooting methodReferences:
| [1] | R. P. Agarwal & Y. M. Chow (1986). Finite difference methods for boundary value prob-lems of differential equations with deviating arguments. Computational & Mathematics with Applications, 12(11, Part A), 1143-1153. https://doi.org/10.1016/0898-1221(86)90018-0. · Zbl 0617.65092 · doi:10.1016/0898-1221(86)90018-0 |
| [2] | A. Andargie & Y. N. Reddy (2013). Parameter fitted scheme for singularly perturbed delay differential equations. International Journal of Applied Science and Engineering, 11(4), 361-373. https://doi.org/10.6703/IJASE.2013.11(4).361. · doi:10.6703/IJASE.2013.11(4).361 |
| [3] | V. L. Bakke & Z. Jackiewicz (1989). The numerical solution of boundary value problems for differential equations with state dependent deviating arguments. Aplikace Matematiky, 34(1), 1-17. http://eudml.org/doc/15560. · Zbl 0669.65065 |
| [4] | Z. Bartoszewski (2010). A new approach to numerical solution of fixed-point problems and its application to delay differential equations. Applied Mathematics and Computation, 215(12), 4320-4331. https://doi.org/10.1016/j.amc.2009.12.058. · Zbl 1191.65059 · doi:10.1016/j.amc.2009.12.058 |
| [5] | J. Biazar & B. Ghanbari (2012). The homotopy perturbation method for solving neutral functional-differential equations with proportional delays. Journal of King Saud University-Science, 24(1), 33-37. https://doi.org/10.1016/j.jksus.2010.07.026. · doi:10.1016/j.jksus.2010.07.026 |
| [6] | X. Chen & L. Wang (2010). The variational iteration method for solving a neutral functional-differential equation with proportional delays. Computers & Mathematics with Applications, 59(8), 2696-2702. https://doi.org/10.1016/j.camwa.2010.01.037. · Zbl 1193.65145 · doi:10.1016/j.camwa.2010.01.037 |
| [7] | P. Chocholaty & L. Slahor (1979). A numerical method to boundary value problems for second order delay differential equations. Numerische Mathematik, 33, 69-75. https://doi. org/10.1007/BF01396496. · Zbl 0432.65046 · doi:10.1007/BF01396496 |
| [8] | C. W. Cryer (1973). The numerical solution of boundary value problems for second order functional differential equations by finite differences. Numerische Mathematik, 20, 288-299. https://doi.org/10.1007/BF01407371. · Zbl 0247.65053 · doi:10.1007/BF01407371 |
| [9] | R. D. Driver (1977). Ordinary and delay differential equations volume 20. Springer-Verlag New York Inc, New York. · Zbl 0374.34001 |
| [10] | N. Jaaffar, Z. Majid & N. Senu (2021). Numerical computation of third order delay differential equations by using direct multistep method. Malaysian Journal of Mathematical Sciences, 15(3), 369-385. · Zbl 1483.65113 |
| [11] | Z. Jackiewicz (1982). Adams methods for neutral functional differential equations. Nu-merische Mathematik, 39, 221-230. https://doi.org/10.1007/BF01408695. · Zbl 0491.65044 · doi:10.1007/BF01408695 |
| [12] | Z. Jackiewicz & E. Lo (2006). Numerical solution of neutral functional differential equations by Adams methods in divided difference form. Journal of Computational and Applied Mathe-matics, 189(1-2), 592-605. https://doi.org/10.1016/j.cam.2005.02.016. · Zbl 1090.65084 · doi:10.1016/j.cam.2005.02.016 |
| [13] | C. Li & C. Zhang (2017). The extended generalized Störmer-Cowell methods for second-order delay boundary value problems. Applied Mathematics and Computation, 294, 87-95. http: //dx.doi.org/10.1016/j.amc.2016.09.006. · Zbl 1411.65085 · doi:10.1016/j.amc.2016.09.006 |
| [14] | Z. A. Majid, P. S. Phang & M. Suleiman (2011). Solving directly two point non linear bound-ary value problems using direct Adams Moulton method. Journal of Mathematics and Statistics, 7(2), 124-128. https://doi.org/10.3844/jmssp.2011.124.128. · doi:10.3844/jmssp.2011.124.128 |
| [15] | L. Muhsen & N. Maan (2016). Lie group analysis of second-order non-linear neutral delay differential equations. Malaysian Journal of Mathematical Sciences, 10(S), 117-129. |
| [16] | K. D. Nevers & K. Schmitt (1971). An application of the shooting method to boundary value problems for second order delay equations. Journal of Mathematical Analysis and Applications, 36, 588-597. · Zbl 0219.34050 |
| [17] | R. Qu & R. P. Agarwal (1998). A subdivision approach to the contruction of approximate so-lutions of boundary value problems with deviating arguments. Computational & Mathematics with Applications, 35(11), 121-135. https://doi.org/10.1016/S0898-1221(98)00089-3. · Zbl 0999.65080 · doi:10.1016/S0898-1221(98)00089-3 |
| [18] | G. W. Reddien & C. C. Travis (1974). Approximation methods for boundary value prob-lems of differential equations with functional arguments. Journal of Mathematical Analysis and Applications, 46(1), 62-74. https://doi.org/10.1016/0022-247X(74)90281-9. · Zbl 0321.65062 · doi:10.1016/0022-247X(74)90281-9 |
| [19] | M. G. Sakar (2017). Numerical solution of neutral functional-differential equations with pro-portional delays. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 7(2), 186-194. https://doi.org/10.11121/ijocta.01.2017.00360. · doi:10.11121/ijocta.01.2017.00360 |
| [20] | H. Y. Seong (2016). Multistep Blocks Methods For Solving Higher Order Delay Differential Equa-tions. PhD thesis, Universiti Putra Malaysia, Malaysia,. |
| [21] | C. L. Sirisha & Y. N. Reddy (2017). Numerical integration of singularly perturbed delay dif-ferential equations using exponential integrating factor. Mathematical Communications, 22(2), 251-264. · Zbl 1383.65070 |
| [22] | C. Tunç (2015). Convergence of solutions of nonlinear neutral differential equations with multiple delays. Boletín de la Sociedad Matemática Mexicana, 21(2), 219-231. https://doi.org/ 10.1007/s40590-014-0050-6. · Zbl 1327.34126 · doi:10.1007/s40590-014-0050-6 |
| [23] | W. Wang, Y. Zhang & S. Li (2009). Stability of continuous Runge-Kutta-type methods for nonlinear neutral delay-differential equations. Applied Mathematical Modelling, 33(8), 3319-3329. https://doi.org/10.1016/j.apm.2008.10.038. · Zbl 1205.65214 · doi:10.1016/j.apm.2008.10.038 |
| [24] | W. S. Wang & S. F. Li (2007). On the one-leg θ-methods for solving nonlinear neutral functional differential equations. Applied Mathematics and Computation, 193(1), 285-301. https://doi.org/10.1016/j.amc.2007.03.064. · Zbl 1193.34156 · doi:10.1016/j.amc.2007.03.064 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
