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The Bubnov-Galerkin method for an abstract quasilinear problem on stationary action. (English. Russian original) Zbl 0863.34065

Differ. Equations 31, No. 7, 1169-1179 (1995); translation from Differ. Uravn. 31, No. 7, 1222-1231 (1995).
The paper deals with the Cauchy problem for the quasilinear second order evolution equation \(Ju''(t)+ Au(t)+ Bu(t)= f(t)\), \(t\in[0,T]\), in a Hilbert space \(H\), where \(u\) and \(f:[0,T]\to H\) are the unknown and given functions respectively, \(J\) and \(A\) are selfadjoint positive definite operators in \(H\), \(B\) is a nonlinear operator from \(H_A\) into \(H\), and \(H_A\) is the energy space generated by \(A\). This problem is an abstract version of initial-boundary value problems in mathematical physics obtained on the basis of the Ostrogradskii-Hamilton variational principle. The author proves the existence and uniqueness of the above Cauchy problem and finds a priori error estimates of its solutions by the Bubnov-Galerkin method by using eigenvectors of some auxiliary operator as a basis. These estimates allow one to state the strong convergence of the sequence of approximate solutions. In conclusion, the Cauchy problem for a quasilinear hyperbolic equation and the IBV problem for an integro-partial differential equation illustrate the obtained results.

MSC:

34G20 Nonlinear differential equations in abstract spaces
65J15 Numerical solutions to equations with nonlinear operators
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65Z05 Applications to the sciences
35L15 Initial value problems for second-order hyperbolic equations
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