Numerical solution of a nonlinear integro-differential equation. (English) Zbl 0488.65074
Keywords:
Galerkin’s method; nonlinear integro-differential equation; error estimate; rate of convergence; numerical experimentsReferences:
| [1] | Barbu, V., Integro-differential equations in Hilbert spaces, An. Stünt. · Zbl 0366.45013 |
| [2] | Ciarlet, P., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0383.65058 |
| [3] | Coleman, B. D.; Gurtin, M. E., Waves in materials with memory II, on the growth and decay of one-dimensional acceleration waves, Arch. Rational Mech. Anal., 19, 239-265 (1965) · Zbl 0244.73017 |
| [4] | Dendy, J. E., Galerkin’s method for some highly nonlinear problems, SIAM J. Numer. Anal., 14, 327-347 (1977) · Zbl 0365.65065 |
| [5] | Glimm, J.; Lax, P. D., Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. Amer. Math. Soc., 101 (1970) · Zbl 0204.11304 |
| [6] | Greenberg, J. M.; MacCamy, R. C.; Mizel, V. J., On the existence, uniqueness and stability of solutions of the equation \(ϱ\) − \(χ_{ tt } = E(χ_x)χ_{ xx } + λχ_{ xxt } \), J. Math. Mech., 17, 707-728 (1968) · Zbl 0157.41003 |
| [7] | Gurtin, M. E.; Pipkin, A. C., A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31, 113-126 (1968) · Zbl 0164.12901 |
| [8] | Klainerman, S., Global existence for nonlinear wave equations, Comm. Pure Appl. Math., 33, 43-101 (1980) · Zbl 0405.35056 |
| [9] | Lax, P. D., Development of singularities of solutions of nonlinear hyperbolic differential equations, J. Math. Phys., 5, 611-613 (1964) · Zbl 0135.15101 |
| [10] | MacCamy, R. C., Existence, uniqueness and stability of \(utt = (∂∂x)(σ(ux) + λ(ux) uxt)\), Indiana Univ. Math. J., 20, 331-338 (1970) |
| [11] | MacCamy, R. C., Stability theorems for a class of functional differential equations, SIAM J. Appl. Math., 30, 557-576 (1976) · Zbl 0346.34059 |
| [12] | MacCamy, R. C., An integro-differential equation with application in heat flow, Quart. Appl. Math., 35, 1-19 (1977) · Zbl 0351.45018 |
| [13] | MacCamy, R. C., A model for one-dimensional, nonlinear viscoelasticity, Quart. Appl. Math., 35, 21-33 (1977) · Zbl 0355.73041 |
| [14] | Matsumara, A., Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with first-order dissipation, Publ. Res. Inst. Math. Sci. Kyoto Univ. Ser. A, 13, 349-379 (1977) · Zbl 0371.35030 |
| [15] | A. Matsumara; A. Matsumara |
| [16] | Neta, B., Finite element approximation of a nonlinear parabolic problem, Comput. Math. Appl., 4, 247-255 (1978) · Zbl 0397.65073 |
| [17] | T. Nishida; T. Nishida |
| [18] | Nohel, J. A., A forced quasilinear wave equation with dissipation, (Fabera, J., Proceedings, EQUADIFF 4. Proceedings, EQUADIFF 4, Prague, August 1977. Proceedings, EQUADIFF 4. Proceedings, EQUADIFF 4, Prague, August 1977, Lecture Notes in Math. No. 703 (1979), Springer-Verlag: Springer-Verlag New York), 318-327 · Zbl 0401.35074 |
| [19] | Staffans, O. J., On a nonlinear hyperbolic Volterra equation, SIAM J. Math. Anal., 11, 793-812 (1980) · Zbl 0464.45010 |
| [20] | Strang, G.; Fix, G. J., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, N. J · Zbl 0278.65116 |
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