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**Numerical schemes for one-dimensional parabolic equations with nonstandard initial condition.**
*(English)*
Zbl 1033.65068

Summary: Parabolic partial differential equations with nonstandard initial condition, feature in the mathematical modelling of many phenomena. While a significant body of knowledge about the theory and numerical methods for parabolic partial differential equations with classical initial condition has been accumulated, not much has been extended to parabolic partial differential equations with nonstandard initial condition.

In this paper finite difference approximations to the solution of one-dimensional parabolic equation with nonstandard initial condition are studied. Several finite difference schemes are presented for solving a parabolic partial differential equation with nonlocal time weighting initial condition. These schemes are based on the forward time centred space explicit formula, the backward time centred space implicit technique, the Crank-Nicolson implicit formula, the Crandall’s implicit scheme and the Saulyev’s (2,2) explicit methods. The results of a numerical experiment are presented.

In this paper finite difference approximations to the solution of one-dimensional parabolic equation with nonstandard initial condition are studied. Several finite difference schemes are presented for solving a parabolic partial differential equation with nonlocal time weighting initial condition. These schemes are based on the forward time centred space explicit formula, the backward time centred space implicit technique, the Crank-Nicolson implicit formula, the Crandall’s implicit scheme and the Saulyev’s (2,2) explicit methods. The results of a numerical experiment are presented.

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35K05 | Heat equation |

### Keywords:

Nonstandard initial condition; Finite difference schemes; Nonlocal time weighting initial condition; Explicit techniques; Implicit methods; Crank-Nicolson implicit formula; numerical experiments
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\textit{M. Dehghan}, Appl. Math. Comput. 147, No. 2, 321--331 (2004; Zbl 1033.65068)

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### References:

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