Numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument. (English) Zbl 1386.34126
Summary: This work is devoted to the numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument in minor terms. The sufficient conditions of the one-valued solvability are established, and the a priori estimate of the solution is obtained. For the numerical solution, the problem studied is reduced to the equivalent boundary value problem for an ordinary linear differential equation of fourth order, for which the finite-difference scheme of second-order approximation was built. The convergence of this scheme to the exact solution is shown under certain conditions of the solvability of the initial problem. To solve the finite-difference problem, the method of five-point marching of schemes is used.
MSC:
| 34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |
Keywords:
equation with deviating argument; a priori estimate of the solution; finite-difference scheme; method of marching for five-point systemsReferences:
| [1] | M. H. Abregov, V. Z. Kanchukoev and M. A. Shardanova, Numerical methods for solving the first kind boundary value problem for a linear second order differential equation with a deviating argument on a symmetric interval, Int. Research J. (2016), no. 5(47), 11-15.; Abregov, M. H.; Kanchukoev, V. Z.; Shardanova, M. A., Numerical methods for solving the first kind boundary value problem for a linear second order differential equation with a deviating argument on a symmetric interval, Int. Research J., 5-47, 11-15 (2016) · Zbl 1386.34126 |
| [2] | M. H. Abregov, V. Z. Kanchukoev and M. A. Shardanova, The first kind boundary value problem for a linear second order differential equation with a deviating argument on a symmetric interval, Int. Research J. (2016), no. 5(47), 6-11.; Abregov, M. H.; Kanchukoev, V. Z.; Shardanova, M. A., The first kind boundary value problem for a linear second order differential equation with a deviating argument on a symmetric interval, Int. Research J., 5-47, 6-11 (2016) · Zbl 1386.34126 |
| [3] | A. B. Mudrov, On the relations of the systems of ordinary differential equations and equations with a delayed agrument, Bull. Novosibirsk State Univ. 7 (2007), no. 2, 52-64.; Mudrov, A. B., On the relations of the systems of ordinary differential equations and equations with a delayed agrument, Bull. Novosibirsk State Univ., 7, 2, 52-64 (2007) · Zbl 1249.34181 |
| [4] | S. B. Norkin, Differential Equations of the Second Order with Retarded Argument, Transl. Math. Monogr. 31, American Mathematical Society, Providence, 1965.; Norkin, S. B., Differential Equations of the Second Order with Retarded Argument (1965) |
| [5] | A. V. Prosolov, Dynamic Models with Delay and Their Applications in Economics and Engineering, LAN Publishing, St. Peterburg, 2010.; Prosolov, A. V., Dynamic Models with Delay and Their Applications in Economics and Engineering (2010) |
| [6] | A. A. Samarskiy, The Theory of Difference Schemes, The Science, Moscow, 1989.; Samarskiy, A. A., The Theory of Difference Schemes (1989) · Zbl 0874.65077 |
| [7] | N. A. Tikhonov, A. B. Vasil’eva and A. G. Sveshnikov, Differential Equations, Fizmatlit, Moscow, 1985.; Tikhonov, N. A.; Vasil’eva, A. B.; Sveshnikov, A. G., Differential Equations (1985) · Zbl 0553.34001 |
| [8] | A. B. Vasil’eva and N. A. Tikhonov, Integral Equations, Fizmatlit, Moscow, 2002.; Vasil’eva, A. B.; Tikhonov, N. A., Integral Equations (2002) · Zbl 1017.45001 |
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.
