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(d) Conclude that the solution to the Cauchy problem for Laplace equation does not depend continuously on the initial data. In other words, the initial value problem (Cauchy problem) for the Laplace equation is not well-posed. It may be noted that a problem involving a PDE is well-posed if the following three properties are satisfied: (i) The solution to the problem exists. (ii) The solution is unique. (iii) The solution depends continuously on the data of the problem. Fortunately, many a physical phenomena give rise to initial or boundary or IBVPS which are well-posed.

(d) Conclude that the solution to the Cauchy problem for Laplace equation does not depend continuously on the initial data. In other words, the initial value problem (Cauchy problem) for the Laplace equation is not well-posed. It may be noted that a problem involving a PDE is well-posed if the following three properties are satisfied: (i) The solution to the problem exists. (ii) The solution is unique. (iii) The solution depends continuously on the data of the problem. Fortunately, many a physical phenomena give rise to initial or boundary or IBVPS which are well-posed.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 10CR
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(d) Conclude that the solution to the Cauchy problem for Laplace equation does not depend continuously on the initial data. In other words, the initial value problem (Cauchy problem) for the Laplace equation is not well-posed. It may be noted that a problem involving a PDE is well-posed if the following three properties are satisfied: (i) The solution to the problem exists. (ii) The solution is unique. (iii) The solution depends continuously on the data of the problem. Fortunately, many a physical phenomena give rise to initial or boundary or IBVPS which are well-posed.
Transcribed Image Text:(d) Conclude that the solution to the Cauchy problem for Laplace equation does not depend continuously on the initial data. In other words, the initial value problem (Cauchy problem) for the Laplace equation is not well-posed. It may be noted that a problem involving a PDE is well-posed if the following three properties are satisfied: (i) The solution to the problem exists. (ii) The solution is unique. (iii) The solution depends continuously on the data of the problem. Fortunately, many a physical phenomena give rise to initial or boundary or IBVPS which are well-posed.
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