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

u(x,t)=∑n=1∞AnXn(x)Tn(t)u open paren x comma t close paren equals sum from n equals 1 to infinity of cap A sub n cap X sub n open paren x close paren cap T sub n open paren t close paren Use the initial condition (e.g., ) to determine the coefficients Ancap A sub n

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

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 (

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)

. 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 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

Since the PDE is linear, any linear combination of your product solutions is also a solution. Express the general solution as an infinite sum :