Methods of Mathematical Physics Integral transformation
Methods of Mathematical Physics Integral Transformation
Integral Transformation a General concept of the integra transformation Fundamental ideas of the method of Integral transformation Fourier Transforms of wave problems a Fourier Transforms of heat problems Fourier Transforms of steady problems Conclusion of this charter
Integral Transformation ◼ General concept of the integral transformation ◼ Fundamental ideas of The method of Integral Transformation ◼ Fourier Transforms of wave problems ◼ Fourier Transforms of heat problems ◼ Fourier Transforms of steady problems ◼ Conclusion of this charter
General Concepts a Integral transformation Definition turn the oragl function into one respect to new coefficients e General form f(x)→F(k)=∫f(×)K(xk)dx General properties linearity:af()+βg(X)→αF(k)+βG(k) ● Inverse transformation:f(×)←→F(k) derivative f(x)→f(×)K(Xk)| bound-∫f(×)Kx(x,k)dx
General Concepts ◼ Integral transformation ◼ Definition ： • Turn the oragl function into one respect to new coefficients • General form • f(x) → F(k) = ∫f(x) K(x,k) dx ◼ General properties： • linearity： f(x) + g(x) → F(k) + G(k) • Inverse transformation：f(x) ←→ F(k) • derivative： f’(x) → f(x) K(x,k)|bound - ∫f(x) Kx(x,k) dx
General Concepts Typical integral transformations F(k)= f(r) dx ■ Fourier 丌 Transfor mation f(x)= F(k)e dk a Laplace F(p)=Lf(x)e-p*dx Transfor O+10 mation f(x) 2ai do-ioo F(p)ep dp
General Concepts ◼ Typical integral transformations： ◼ Fourier Transfor mation f x F k e dk F k f x e dx ikx ikx − − − = = ( ) ( ) ( ) 2 1 ( ) ◼ Laplace Transfor mation + − − = = i i p x p x F p e dp i f x F p f x e dx ( ) 2 1 ( ) ( ) ( ) 0
Fundamental idea ■ Programmer: Turn the original equation into an ordinary differential equation Solve the ordinary differential equation a Charge the solution of the ordinary differential equation into the solution of the original a Applying range: Fourier Transformation Unbounded region aplace transformation ·Semi- unbounded region
Fundamental Idea ◼ Programmer： ◼ Turn the original equation into an ordinary differential equation. ◼ Solve the ordinary differential equation. ◼ Charge the solution of the ordinary differential equation into the solution of the original. ◼ Applying range： ◼ Fourier Transformation ： • Unbounded region ◼ Laplace Transformation ： • Semi- unbounded region
Fourier transform for the wave problem Solving the wave problems l(x,t)→>U(k,t) a2u=0 q(x)→>Φ(k) U +aku=0 y(x)→>Y(k) u 1=09( l12=0=/(x) U=Φ coskat expat y(s)ds +psin kat 2a Jx-at
Fourier Transform for the wave problem = = + = = = 0 0 2 2 | | 0 t t t tt U U U a k U + − + = + + − x at x at s ds a u x at x at ( ) 2 1 [ ( ) ( )] 2 1 Solving the wave problems = = − = = = | ( ) | ( ) 0 0 0 2 u x u x u a u t t t tt xx kat U kat ka sin cos + 1 = ( ) ( ) ( ) ( ) ( , ) ( , ) x k x k u x t U k t → → → u = U k t e dk ikx ( , )
Fourier transform for the heat problems Solving the heat problems l(x,)→>U/(k,t) 4-aZ=0 q(x)→>d(k) U+aku=0 Ulo=Φ 1=0 u=U(k, t)e dk u=vs)ds expl 4at U=dexp(-k'a't
Fourier Transform for the heat problems = + = =0 2 2 | 0 t t U U a k U a t u s ds a t x s 2 exp[ ] ( ) 2 2 4 ( − ) − = Solving the heat problems = − = = | ( ) 0 0 2 u x u a u t t xx exp( ) 2 2 U = −k a t ( ) ( ) ( , ) ( , ) x k u x t U k t → → u = U k t e dk ikx ( , )
Fourier transform for the steady problems Solving the steady problems urr+uw=0, y>0 l(x,y)→U(k,y) k2U=0 0(x)→>d(k) l=0=0(x) U y=0 y→> u=U(k, y)e dk ds l=0(s) Jy U=exp(k y Tx-s)+y
Fourier Transform for the steady problems = = − = → = | 0 | 0 0 2 y y yy U U U k U [( ) ] ( ) 2 2 x s y yds u s − + = Solving the steady problems = = + = → = | 0 | ( ) 0, 0 0 y y xx yy u u x u u y U = exp(− | k | y) ( ) ( ) ( , ) ( , ) x k u x y U k y → → u = U k y e dk ikx ( , )
Conclusion for this charter a In principle, Fourier Transform can be applied to any unbounded problems The steady problem a The heat problem a The wave problem a The key to solving the Eg. by use of Fourier transform method is inverse transfrom a generally the results from Fourier Transform method are integral formulas
Conclusion for this charter ◼ In principle, Fourier Transform can be applied to any unbounded problems ◼ The steady problem ◼ The heat problem ◼ The wave problem ◼ The key to solving the Eq. by use of Fourier Transform method is inverse transfrom ◼ Generally, the results from Fourier Transform method are integral formulas