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Spline methods for the solution of fourth-order parabolic partial differential equations. (English) Zbl 1082.65564

Summary: A fourth-order nonhomogeneous parabolic partial differential equation, that governs the behaviour of a vibrating beam, is solved by using a new three level method based on parametric quintic spline in space and finite difference discretization in time. Stability analysis of the method has been carried out. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and nonhomogeneous cases can be derived from our method. We also obtain two new high accuracy schemes of \(O(k^{4}, h^{6})\) and \(O(k^{4}, h^{8})\) and two new schemes which are analogues of Jain’s formula for the nonhomogeneous case. Comparison of our results with those of some known methods show the superiority of the present approach.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K30 Initial value problems for higher-order parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
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