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Calculus and Beyond Homework Help
Method of Characteristics, PDE, Jacobian condition Q
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[QUOTE="binbagsss, post: 4966424, member: 252335"] Thanks for your reply. I think understand the majority of it. Although I fear I may be missing a few things as I don't fully undersand entirely why we need the characteristics to not be tangential to the curve where the initial conditions are prescribed. From what I understand, when initial conditions are not parallel to the level curves there is no issue: ##J ≠ 0##. When they are parallel ##J ≠ 0## or ## J=0 ##. If the initial conditions are constant on the characteristic we either get infinite or a unique solution, depending upon whether ##J ≠ 0## or ## J=0 ## intially. (Looking at the definition you gave for a characteristic, this means the initial data must be a characteristic curve).If not constant, no solution. I'm a little confused about using the terms 'tangential' and 'parallel' because if tangential we have ##J=0## and in the above discussion we only have parallel sometimes when ##J=0## . So tangential to the intial curve must imply two things: 1)the initial curve and the characterisic are parallel 2) the initial curve is constant on the char I think then, the next step in deriving the intial condition in my original post is to satisfy the 2 criteria listed just above. Criteria 1: ##a_{0}=k u_{0}##, ##b_{0}=k u_{0}##. k some constant Criteria 2: ## \frac{ \partial u_{0}}{ \partial t } =0=\frac{ \partial u_{0}}{ \partial x }a + \frac{ \partial u_{0}}{ \partial y }b ## But these 2 equations didnt help me get to the condition Have I understood correctly and are my thoughts on the right or wrong track? Thanks in advance. [/QUOTE]
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Method of Characteristics, PDE, Jacobian condition Q
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