. This often involves calculating a Fourier Sine or Cosine Series for the function using orthogonality integrals . For a sine series on , the formula is:
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 Partial Differential Equations with Fourier Ser...
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 Fourier Series are then used to combine these
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 ✅ Because they depend on different variables but are
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
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 (