. This often involves calculating a Fourier Sine or Cosine Series for the function using orthogonality integrals . For a sine series on , the formula is:
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
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 ✅ Partial Differential Equations with Fourier Ser...
Since the PDE is linear, any linear combination of your product solutions is also a solution. Express the general solution as an infinite sum :
Plug the calculated coefficients back into your general series solution. For the Heat Equation with zero-temperature boundary conditions, the solution typically looks like: Express the general solution as an infinite sum
), which you solve using the given boundary conditions (like ) to find specific values for and their corresponding eigenfunctions .
). The spatial ODE is typically an eigenvalue problem (e.g., The spatial ODE is typically an eigenvalue problem (e
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