Partial Differential Equations - Variable Seperable Solutions

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SUMMARY

The discussion centers on the proof of the existence of variable separable solutions in linear partial differential equations (PDEs). It is established that nearly all functions can be expressed as linear combinations of these solutions, particularly through infinite sums or integrals. The conversation highlights that any analytic function of two variables can be represented as a Taylor series, while periodic functions can be decomposed into sine and cosine products. The broader implications of this include the use of Fourier Transforms to expand the solution space of linear PDEs.

PREREQUISITES
  • Understanding of linear partial differential equations (PDEs)
  • Familiarity with variable separation technique
  • Knowledge of Taylor series and Fourier Transforms
  • Basic concepts of function spaces in mathematics
NEXT STEPS
  • Study the method of variable separation in linear PDEs
  • Explore Taylor series expansions for multivariable functions
  • Learn about Fourier Transforms and their applications in PDEs
  • Investigate the properties of function spaces relevant to linear PDE solutions
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Mathematicians, physics students, and engineers interested in solving linear partial differential equations and understanding the theoretical foundations of variable separable solutions.

TZW85
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Hi there,

Does anyone know of a proof of why, in partial DEs, one can assume the existence of variable seperable solutions, then take the linear combination of all of them to be the general solution? Why can't there be any other funny solutions that fall outside the space spanned by these variable seperable ones?
 
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TZW85 said:
Hi there,

Does anyone know of a proof of why, in partial DEs, one can assume the existence of variable seperable solutions, then take the linear combination of all of them to be the general solution? Why can't there be any other funny solutions that fall outside the space spanned by these variable seperable ones?
Because (almost) any function can be written that way! It's not a matter of "no solutions that fall outside the space"- there are (almost)no functions that fall outside the space- if you allow infinite sums. For example, any analytic function of x and y can be written as a Taylor series in x and y- a sum powers of x and powers of y. Any periodic function of x and y, even if not continuous, can be written as a sum of products of sin or cos of x times sin or cos of y. If you allow integrals rather than sums of such functions, such as Fourier Transforms, the space of all functions that can be written in that form is much larger.
 
We are talking about properties of solutions to LINEAR partial DE's here, I hope.
 
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