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The Laplace equation $\nabla^2 \psi = 0$ is a linear differential equation. Now note that if $\phi$ is real, then so is $\nabla^2 \phi$. Moreover, by the linearity of the equation, if $\phi$ is real, then $i\phi$ is pure imaginary, and so is $\nabla^2(i\phi) = i \nabla^2(\phi)$. Okay, back to your situation. Let's say the solution is $\phi_1 + i\phi_2$ for ...


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I just found that second volume of Gas Dynamics by Zucrow wholly deals with the use of Method of Characteristics to solve various kinds of flow problems. I hope it will solve my problem.


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This is an answer by an experimentalist who had been fitting data with mathematical models since 1968. When fitting data one goes to the simplest mathematical models. When the data display variations in time and space the Fourier expansion is extremely useful because it gives the frequencies and amplitudes that will fit a periodic data set. One gets as ...


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In practice, the simple answer is that it works when it does. You need to test an unkown situation at hand, by making the assumption of a separated solution $f(x,\,y,\,\cdots)=X(x)\,Y(y)\,\cdots$ and seeing what comes from that. Pretty much all the equations you will need for electromagnetism have been tested for separation in co-ordinates that reflect ...


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I see that your goal is to design a supersonic nozzle. I will answer your questions with that in mind. I want to know what it is It is a method for solving hyperbolic partial differential equations. In the case of supersonic flow, the method of characteristics defines paths through the flow for which certain quantities are known (or easily ...



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