To solve Partial Differential Equations (PDEs) like the Heat Equation or the Wave Equation , you use the method of separation of variables to turn a multivariable equation into several Ordinary Differential Equations (ODEs). Fourier Series are then used to combine these individual solutions to satisfy the initial and boundary conditions of the original problem. Assume the solution can be written as a product of two independent functions, . Substitute this into the PDE to isolate all terms on one side and all

. This often involves calculating a Fourier Sine or Cosine Series for the function using orthogonality integrals . For a sine series on , the formula is:

To solve a PDE with Fourier Series, you break the equation into independent parts, solve for the specific patterns (eigenfunctions) that fit the boundaries, and then sum those patterns to match the initial starting state. 3. Fourier Series in Partial Differential Equations (PDEs)

Plug the calculated coefficients back into your general series solution. For the Heat Equation with zero-temperature boundary conditions, the solution typically looks like:

u(x,t)=∑n=1∞Ansin(nπxL)e−k(nπL)2tu open paren x comma t close paren equals sum from n equals 1 to infinity of cap A sub n sine open paren the fraction with numerator n pi x and denominator cap L end-fraction close paren e raised to the exponent negative k open paren the fraction with numerator n pi and denominator cap L end-fraction close paren squared t end-exponent ✅

terms on the other. Because they depend on different variables but are equal, both sides must equal a constant, typically denoted as −λnegative lambda This yields two separate ODEs: one for space ( ) and one for time (