Rothe’s method for a telegraph equation with integral conditions. (English) Zbl 1171.35306

Guezane-Lakoud, A.; Belakroum, D.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 11, 3842-3853 (2009).

The paper deals with the telegraph equation \[ \tau\frac{\partial^2 v}{\partial t^2}+a\frac{\partial v}{\partial t}-b\frac{\partial^2 v}{\partial x^2}=f(x,t,v),\quad (x,t)\in (0,1)\times [0,T], \] subject to the initial conditions \[ v(x,0)=v_0(x),\quad \frac{\partial v}{\partial t}(x,0)=v_1(x), \] the Neumann condition \[ \frac{\partial v}{\partial x}(0,t)=G(t) \] and the integral condition \[ \int_0^1v(x,t)\,dx=E(t), \] where \(f,\,v_1,\,v_0,\,G,\,E\) are given functions which verify some assumptions, \(T,\,\tau,\,b\) are positive constants and \(a\geq 1\). By using the Rothe time discretization method, the authors prove the existence and uniqueness of the weak solutions of the above problem.

Reviewer: Rodica Luca Tudorache (Iaşi)

MSC:

35A35	Theoretical approximation in context of PDEs
35L70	Second-order nonlinear hyperbolic equations
35D05	Existence of generalized solutions of PDE (MSC2000)
35L20	Initial-boundary value problems for second-order hyperbolic equations

Keywords:

Rothe time discretization; existence and uniqueness of the weak solutions; a priori estimate; telegraph equation; weak solution

References:

[1]	Bahuguna, D.; Shukla, R.K., Partial functional differential equations and applications to population dynamics, Nonlinear dyn. syst. theory, 5, 4, 345-356, (2005) · Zbl 1111.35095
[2]	Bahuguna, D.; Abbas, S.; Dabas, J., Partial functional differential equation with an integral condition and applications to population dynamics, Nonlinear anal., (2007) · Zbl 1155.35478
[3]	Beilin, S.A., Existence of solutions for one dimensional wave equations with nonlocal conditions, Electron. J. differential equations, 76, 1-8, (2001) · Zbl 0994.35078
[4]	Bouziani, A.; Merazga, N., Solution to a semilinear pseudoparabolic problem with integral conditions, Electron. J. differential equations, 115, 1-18, (2006), N · Zbl 1112.35115
[5]	Bouziani, A.; Merazga, N., Rothe time-discretization method applied to a quasilinear wave equation subject to integral conditions, Adv. difference equations, 3, 211-235, (2001) · Zbl 1077.35097
[6]	Cannon, J.R., The solution of the heat equation subject to the specification of energy, Quart. appl. math., 21, 155-160, (1963) · Zbl 0173.38404
[7]	Ionkin, N.A., Solutions of boundary value problem in heat conduction theory with nonlocal boundary conditions, Differ. uravn., 13, 294-304, (1977)
[8]	Kacur, J., Method of rothe in evolution, (1985), Teubner Leipzig · Zbl 0582.65084
[9]	J. Kacur, R. van Keer, On numerical method for a class of parabolic problems in composite media, Belgian Nat. Sci. Research Found., 1992, Preprint · Zbl 0837.65102
[10]	Kartsatos, A.G.; Liu, X., On the construction and the convergence of the method of lines for quasi-nonlinear functional evolutions in general Banach spaces, Nonlinear anal., 29, 4, 385-414, (1997) · Zbl 0881.34083
[11]	Merazga; Bouziani, A., On a time-discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function space, Nonlinear anal., 66, 3, 604-623, (2007) · Zbl 1105.35044
[12]	Pulkina, L.S., A non-local problem with integral conditions for hyperbolic equations, Electron. J. differential equations, 45, 1-6, (1999) · Zbl 0935.35027