Advertisement

Approximation Methods for Initial Value Problems in Partial Differential Equations

  • H.-J. Reinhardt
Part of the Applied Mathematical Sciences book series (AMS, volume 57)

Abstract

We begin this rather extensive chapter by presenting several numerical methods for solving the heat equation and the wave equation (Section 4.1 to 4.3), which are typical examples of parabolic and hyperbolic problems, respectively. The methods we discuss comprise not only finite-difference methods but also Galerkin methods; our methods are either explicit or implicit and include so-called multilevel (more precisely, three-level) methods. In Section 4.4, we present finite-difference and Galerkin methods for approximating various classes of nonlinear initial value problems and discuss the solvability of the associated systems of nonlinear equations. Finally, we show in Section 4.5 how the problems considered in the previous sections — along with their approximating equations — can be viewed as operator equations in appropriate function spaces.

Keywords

Variational Formulation Heat Equation Galerkin Method Dirichlet Boundary Condition Homogeneous Dirichlet Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ames (1977), Ansorge (1978), Ansorge & Haas (1970), Aubin (1972,1979), Babuska & Aziz (1972)*, Babuska, Prager & Vitasek (1966), Baker (1976)*, Böhmer (1974), Collatz (1966), Dupont (1973)*, Fairweather (1978), Forsythe & Wasow (1960), Gladwell & Wait (1979), Gottlieb & Orszag (1977), Hartman (1964), Hille & Phillips (1957), van der Houwen (1968,1977), Isaacson & Keller (1966), John (1967,1982), Kantorovich & Akilov (1964), Kato (1966), Lions & Magenes (1972), Marchuk (1975), Meis & Marcowitz (1981), Mikhlin & Smolitskiy (1967), Mitchell (1969), Mitchell & Griffiths (1980), Mitchell & Wait (1977), Ortega & Rheinboldt (1970), Parter (1980)*, Richtmyer & Morton (1967), Thomée & Wahlbin (1975)*, Törnig (1979), Trotter (1958)*, Varga (1962), Walter (1976), Yanenko (1971).Google Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • H.-J. Reinhardt
    • 1
  1. 1.Fachbereich MathematikJohann-Wolfgang-Goethe-Universität6000 Frankfurt MainFederal Republic of Germany

Personalised recommendations