Separate a Hamilton-Jacobi PDE by sum.
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The following is an example of a PDE that does not separate by sum (unless E=0).
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The Helmholtz equation in two variables separates by sum and by product.
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A nonlinear PDE system with two unknowns u, v, of three independent variables x, y, z - invoke the declare facility to avoid redundancy in the display of the output (derivatives are displayed as indexed objects):
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As in the case of a single PDE, for PDE systems, unless indicated otherwise, the test is regarding separability by sum
Note that these separability test results are obtained only considering related integrability conditions, not actually separating the variables. Regarding separation by product,
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This result means there exists no separable solution unless the three equations above are zero, that is: there is no solution really involving the product of three functions respectively depending on x, y, z. That doesn't mean there exists no separable solution where one or both of the unknowns {u(x, y, z) v(x, y, z)} depend on less variables (i.e. is a constant with respect to one or more of x, y, z). For example, consider the restriction
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Evaluate remain at this restriction
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So with this restriction the separability conditions are satisfied. In any case the solutions separable by `+` and by `*` can be computed with pdsolve using the HINT option; for this particular system only the solutions separable by `+` have u and v depending on the three variables
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