Exact solutions and numerical approximations of mixed problems for the wave equation with delay. (English) Zbl 1309.35045

Summary: This work deals with the construction of exact and continuous numerical solutions of mixed problems for the wave equation with delay \[ u_{tt}(t,x)=a^2u_{xx}(t,x)+bu(t-\tau,x), \quad t>\tau, \quad 0\leq x\leq l. \] Using the method of separation of variables, and obtaining explicit expressions for the solution of a general initial value problem for a second order delay differential equation, an exact infinite series solution is derived. Bounds on the truncation errors, when this series is approximated by finite sums, are given, thus providing constructive continuous numerical solutions with prescribed accuracy in bounded domains.


35L20 Initial-boundary value problems for second-order hyperbolic equations
35C05 Solutions to PDEs in closed form
Full Text: DOI


[1] Kolmanovskii, V.; Myshkis, A., Introduction to the theory and applications of functional differential equations, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0917.34001
[2] Wu, J., Theory and applications of partial functional differential equations, (1996), Springer-Verlag New York
[3] He, M.; Liu, A., The oscillation of hyperbolic functional differential equations, Appl. math. comput., 142, 205-224, (2003) · Zbl 1028.35151
[4] Wang, J.; Meng, F.; Liu, S., Integral average method for oscillation of second order partial differential equations with delays, Appl. math. comput., 187, 815-823, (2007) · Zbl 1119.35108
[5] Fabrizio, M.; Morro, A., Mathematical problems in linear viscoelasticity, SIAM studies in applied mathematics, vol. 12, (1992), SIAM Philadelphia · Zbl 0753.73003
[6] N. Raskin, Y. Halevi, Control of flexible structures governed by the wave equation, in: Proceedings of the American Control Conference, Arlington, VA, 2001, pp. 2486-2491.
[7] Nicaise, S.; Pignotti, C., Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. control optim., 45, 5, 1561-1585, (2006) · Zbl 1180.35095
[8] Kucuk, I.; Sadek, I.; Yilmaz, Y., Active control of a smart beam with time delay by Legendre wavelets, Appl. math. comput., 218, 8968-8977, (2012) · Zbl 1245.74032
[9] Fridman, E.; Orlov, Y., Exponential stability of linear distributed parameter systems with time-varying delays, Automatica, 45, 194-201, (2009) · Zbl 1154.93404
[10] Farlow, S.J., Partial differential equations for scientists and engineers, (1993), Dover New York · Zbl 0851.35001
[11] Folland, G.B., Fourier analysis and its applications, (1992), Wadsworth & Brooks Pacific Groove · Zbl 0371.35008
[12] Scott, E.J., On a class of linear partial differential equations with retarded argument in time, Bul. inst. politeh. iaşi, 15, 99-103, (1969) · Zbl 0189.10501
[13] Wiener, J.; Debnath, L., A wave equation with discontinuous time delay, Int. J. math. math. sci., 20, 781-788, (1992) · Zbl 0766.35024
[14] Martín, J.A.; Rodríguez, F.; Company, R., Analytic solution of mixed problems for the generalized diffusion equation with delay, Math. comput. model., 40, 361-369, (2004) · Zbl 1062.35157
[15] Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0115.30102
[16] El’sgol’ts, L.E.; Norkin, S.B., Introduction to the theory and application of differential equations with deviating arguments, (1973), Academic Press New York · Zbl 0287.34073
[17] Andrews, G.E.; Askey, R.; Roy, R., Special functions, (1999), Cambridge University Press Cambridge
[18] Apostol, T.M., Mathematical analysis, (1957), Addison-Wesley Reading · Zbl 0126.28202
[19] Jódar, L.; Almenar, P., Accurate continuous numerical solutions of time dependent mixed partial differential problems, Comput. math. appl., 32, 5-19, (1996) · Zbl 0857.65097
[20] Jódar, L.; Posso, A.E., Analytic numerical approximation with a priori error bound for the wave equation with time dependent coefficient, Math. comput. model., 29, 1-14, (1999) · Zbl 0992.65109
[21] Reyes, E.; Rodríguez, F.; Martín, J.A., Analytic-numerical solutions of diffusion mathematical models with delays, Comput. math. appl., 56, 743-753, (2008) · Zbl 1155.65387
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