Numerical methods for ordinary differential systems: the initial value problem
Numerical methods for ordinary differential systems: the initial value problem
Sideways heat equation and wavelets
Modelling 94 Proceedings of the 1994 international symposium on Mathematical modelling and computational methods
An introduction to the mathematical theory of inverse problems
An introduction to the mathematical theory of inverse problems
Regularized solution of the Cauchy problem for the Laplace equation using Meyer wavelets
Mathematical and Computer Modelling: An International Journal
Mathematics and Computers in Simulation
A regularization method for a Cauchy problem of the Helmholtz equation
Journal of Computational and Applied Mathematics
Mathematical and Computer Modelling: An International Journal
Hi-index | 7.29 |
We consider the Cauchy problem for the Laplace equation in the half plane x 0, y ∈ R where the Cauchy data is given at x = 0 and the solution is sought in the interval 0 x ≤ 1. The problem is ill-posed: the solution (if it exists) does not depend continuously on the data. In order to solve the problem numerically, it is necessary to modify the equation so that a bound on the solution is imposed. We study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the solution of the Cauchy problem for the Laplace equation, and it is shown that when the data error tends to zero, the error in the approximate solution tends to zero logarithmically. Numerical implementation is considered and a simple example is given.