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 ✅

An=2L∫0Lf(x)sin(nπxL)dxcap A sub n equals the fraction with numerator 2 and denominator cap L end-fraction integral from 0 to cap L of f of x sine open paren the fraction with numerator n pi x and denominator cap L end-fraction close paren d x

). The spatial ODE is typically an eigenvalue problem (e.g.,

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