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High accuracy difference formulae for a fourth order quasi-linear parabolic initial boundary value problem of first kind. (English) Zbl 1026.65069

Summary: New three level implicit finite difference methods of \(O(k^2+ h^2)\) and \(O(k^3+ h^4)\) are proposed for the numerical solution of fourth-order quasi-linear parabolic partial differential equations in one space variable, where \(k> 0\) and \(h> 0\) are grid sizes in time and space coordinates respectively. In both cases, we use only nine grid points. The numerical solution of \(\partial u/\partial x\) is obtained as a by-product of the method. The characteristic equation for a model problem is established. Application to a linear singular equation is also discussed in detail. Four examples illustrate the utility of the new difference methods.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
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References:

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[3] DOI: 10.1093/comjnl/8.3.280 · Zbl 0134.33006 · doi:10.1093/comjnl/8.3.280
[4] DOI: 10.1090/S0025-5718-1967-0221785-2 · doi:10.1090/S0025-5718-1967-0221785-2
[5] DOI: 10.1002/nme.1620100614 · Zbl 0345.65047 · doi:10.1002/nme.1620100614
[6] DOI: 10.1016/0045-7825(83)90054-3 · Zbl 0509.65044 · doi:10.1016/0045-7825(83)90054-3
[7] DOI: 10.1080/00207169108804004 · Zbl 0736.65061 · doi:10.1080/00207169108804004
[8] DOI: 10.1016/S0377-0427(99)00202-2 · Zbl 0963.65083 · doi:10.1016/S0377-0427(99)00202-2
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