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
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 ✅
), which you solve using the given boundary conditions (like ) to find specific values for and their corresponding eigenfunctions . Partial Differential Equations with Fourier Ser...
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 (
). The spatial ODE is typically an eigenvalue problem (e.g., To solve Partial Differential Equations (PDEs) like the
so when we get to that point I we'll explain all of these things one after the other but here I'm just trying to give an overview. YouTube·Emmanuel Jesuyon Dansu Heat Equation and Fourier Series
Plug the calculated coefficients back into your general series solution. For the Heat Equation with zero-temperature boundary conditions, the solution typically looks like: Substitute this into the PDE to isolate all
Since the PDE is linear, any linear combination of your product solutions is also a solution. Express the general solution as an infinite sum :