【Mathematical equation】Separation of variables method

2 Separation of variables

2.0 Ordinary Differential Equations

2.0.1 Homogeneous & Inhomogeneous Equations

• Homogeneous equation $Y^{\prime }(x)=P\left(x\right)y\left(x\right)$ separate variables and integrate on both sides
  • $y\left(x\right)=This{\_}^{\int P(x)dx}$
  • P(x) is constant $Y^{\prime }\left(x\right)=my(x)\to y\left(x\right)=This{\_}^{mx}$
• Inhomogeneous equation $Y^{\prime }(x)=P(x)y(x)+Q\left(x\right)$ constant variation method
  • $y\left(x\right)={and}^{\int P(x)dx}(\int Q(x)\cdot {and}^{-\int P(x)dx}dx+C)$

2.0.2 Second-order homogeneous ordinary differential equations with constant coefficients

$a_{}{Y}^{\prime \prime }(x)+a_{}{Y}^{\prime }(x)+a_{}y\left(x\right)=0$

• Special Operation: $a_{}{r}^2+a_{}r+a_{}=0$ , r eigenroot

• 2 real roots $r_{}\ne r_{}$: 通解 $y\left(x\right)=But{you}^{r_{}x}+B{e}^{r_{}x}$

• 1 real root $r_{}=r_{}$: 通解 $y\left(x\right)=(Ax+B)e^{rx}$

• 复根$r_{}=a+\beta i,r_{}=a-\beta i$ : 通解$y\left(x\right)={and}^{\alpha x}(A\cos \beta x+B\sin \beta x)$
  

• $r^2{R}^{"}\left(p\right)+\rho R^{\prime }\left(p\right)-\lambda R\left(\rho \right)=0$

  • $r={and}^x$

2.1 Eigenvalue problem

2.1.1 Common eigenvalue problems

• Ordinary Differential Equation $X^{\prime \prime }(x)+\lambda X(x)=0$

  • $l<0$, $X(x)=But{you}^{\sqrt{-l}x}+B{e}^{-\sqrt{-l}x}$
  • $l=0$, $X\left(x\right)=Ax+B$
  • $l>0$ ,$X\left(x\right)=A\cos \sqrt{l}x+B\sin \sqrt{l}x$

• contains the constant λ \lambda to be determinedThe eigenvalue problem of λ,$\lambda$$\lambda$ eigenvalue,$X\left(x\right)$ characteristic function

• $\{\begin{array}{c}{X}^{\prime \prime }(x)+\lambda X(x)=0\\ X(0)=0,X(l)=\end{array}$

  1. $l<0$ , A=B=0, trivial solution
  2. $l=0$ , A=B=0, trivial solution
  3. $l>0$ ,$\{\begin{array}{c}{l}_n=(\frac{n\pi }{l}{)}^2\\ X_n(x)=B_n\sin \frac{n\pi }{l}x,n=1,2,3\end{array}$
     1. ${\sin \frac{n\pi }{l}x}_{n=1}^{}$
     2. $C_n=\frac{}{l}{\int }_0^{l}f(x)\sin \frac{n\pi }{l}xdx$

2.1.2 Theory on the Eigenvalue Problem

• SLequationddx $\frac{d}{dx}(k(x)\frac{d}{dx})-q(x)y(x)+\lambda \rho (x)y(x)=0,x\in (a,b)$
  • Eigenvalues: Minor Boundary Condition, Periodicity, Natural Boundary Condition
  • $k(x)=p\left(x\right)=1,q(x)=0$: $Y^{\prime \prime }(x)+\lambda y(x)=0$
  • $k(x)=p\left(x\right)=x,q(x)=\frac{n^{}}{x}$: $x^2{y}^{\prime \prime }(x)+x{y}^{\prime }(x)+(\lambda x^2-n^2)and\left(x\right)=0$
  • $k\left(x\right)=1-x^2,p\left(x\right)=1,q\left(x\right)=0$: $(1-x{)}^2{y}^{\prime \prime }(x)-2x{y}^{\prime }(x)+\lambda y(x)=0$
  1. There are countable real eigenvalues ​​-> monotonically increasing sequence $0\le l_{}\le l_{}\le ...\le l_n\le$
     countable eigenfunctions$\{{and}_x(x)\},n=1,2,\mathrm{3...}$
  2. All eigenvalues ​​are non-negative $l_n\ge 0,n=1,2,\mathrm{3...}$
  3. Characteristic function system $\{{and}_n(x){\}}_{n=1}^{n=\infty }$是$L_r^{}[a,b]$ on the weight function$\rho (x)$的正交系${\int }_a^{b}p\left(x\right)y_n\left(x\right)y_m(x)dx=\{\begin{array}{cc}0& n\ne m\\ ||y_n|{|}_2^{}& n=m\end{array}$
  4. Characteristic function system $\{{and}_n(x){\}}_{n=1}^{n=\infty }$是$L_r^{}[a,b]$ on the weight functionρ ( x ) \rho(x)The complete system of$\rho$$($$x$$)$

2.1.3Matlab

• Second Order Ordinary Differential Equation
  Find $\frac{d^2}{d{x}^2}+Y=1-\frac{x^{}}{Pi}$General solution

  • y=dsolve('D2y+y=1-x^2/pi','x')
    

  求$\{\begin{array}{l}\frac{d^2}{d{x}^2}+Y=1-\frac{x^{}}{Pi}\\ and\left(0\right)=0.2,Y^{\prime }\left(0\right)=\end{array}$special solution, drawing

  • y=dsolve('D2y+y=1-x^2/pi','y(0)=0.2,Dy(0)=0.5','x')
    ezplot(y),aixs([-3 3 -0.5 2])
    

• 常微分方程组$\{\begin{array}{l}\frac{of}{dt}=3u-2v\\ \frac{d}{dt}+in=2\end{array}$
  1. Seek a general solution

  • [u,v]=dsolve('Du=3*u-2*v','Dv+v=2*u')
    

  2. To satisfy the initial conditions $in\left(0\right)=1,in\left(0\right)=0$ special solution

  • [u,v]=dsolve('Du=3*u-2*v','Dv+v=2*u','u(0)=1,v(0)=0','t')
    

2.2 One-dimensional wave equation, one-dimensional heat equation solution

2.2.1 One-dimensional wave equation

{ ∂ 2 u ∂ t 2 = a 2 ∂ 2 u ∂ x 2 0 < x < l , t > 0 u ∣ x = 0 = u ∣ x = l = 0 u ∣ t = 0 = ϕ ( x ) , ∂ u ∂ t ∣ t = 0 = ψ ( x ) \left\{\begin{array}{l} \frac{\partial^{2} \boldsymbol{u}}{\partial \boldsymbol{t}^{2}}=\boldsymbol{a}^{2} \frac{\partial^{2} \boldsymbol{u}}{\partial \boldsymbol{x}^{2}} \quad 0<\boldsymbol{x}<\boldsymbol{l}, \boldsymbol{t}>0 \\ \left.\boldsymbol{u}\right|_{\boldsymbol{x}=0}=\left.\boldsymbol{u}\right|_{\boldsymbol{x}=l}=0 \\ \left.\boldsymbol{u}\right|_{t=0}=\phi(\boldsymbol{x}),\left.\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{t}}\right|_{t=0}=\psi(\boldsymbol{x}) \end{array}\right. ⎩ ⎨ ⎧∂ t2∂2 and=a2∂ x2∂2 and0<x<l,t>0u ∣x=0​=u ∣x=l​=0u ∣t=0​=ϕ ( x ) ,∂ t∂ u∣ ∣t=0​=ψ ( x )

• ${in}_n(x,t)$ is sinusoidal at any time
  • $t=t_{}$ ${in}_n(x,t_{})$ is a sinusoid whose amplitude varies with time
  • $x=x_{}$ ${in}_n(x_{},t)$ Simple harmonic vibration
• Arbitrarily determined time ${in}_n(x,t)=A_n^{}\sin \frac{n\pi }{l}x$
  • $n+1$ zero,$n$个极值点,${in}_{},{in}_{},{in}_{}$is a series of standing waves

2.2.2 One-dimensional heat equation solution

2.3 The solution method of Laplace equation definite solution problem

2.3.1 Laplace in Cartesian Coordinate System

2.3.2 The solution of Laplace definite solution problem in two-dimensional circular domain

1. 定解问题$\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}=0$
2. Rationalization $\{\begin{array}{l}x=r\cos i\\ Y=r\sin i\end{array}0\le i\le 2p.m$
3. $\frac{\partial }{\partial x}=\frac{\partial }{\partial \rho }\frac{\partial }{\partial x}+\frac{\partial }{\partial \theta }\frac{\partial }{\partial x}=\mathbf{v}$代入得 $ \begin{array}{l}\frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho \frac{\partial \boldsymbol{u}}{\partial \rho}\right)+\frac{1}{\rho^{2}} \frac{\partial^{2} \boldsymbol{u}}{\partial \theta^{2}}=0 \\frac{\partial^{2} \boldsymbol{u}}{\partial \rho^{2}}+\frac{1}{\rho} \frac{\partial \boldsymbol{u}}{\partial \rho}+\frac{1}{\rho^{2}} \frac{\partial^{2} \boldsymbol{u}}{\partial \theta^{2}}=0\end{array} $

2.4 Inhomogeneous equations

波方程 { ∂ 2 u ∂ t 2 = a 2 ∂ 2 u ∂ x 2 + f ( x , t ) , 0 < x < l , t > 0 u ∣ x = 0 = u ∣ x = l = 0 u ∣ t = 0 = ϕ ( x ) , ∂ u ∂ t ∣ t = 0 = Ψ ( x ) \left\{\begin{array}{l}\frac{\partial^{2} \boldsymbol{u}}{\partial \boldsymbol{t}^{2}}=\boldsymbol{a}^{2} \frac{\partial^{2} \boldsymbol{u}}{\partial \boldsymbol{x}^{2}}+\boldsymbol{f}(\boldsymbol{x}, \boldsymbol{t}), \quad 0<\boldsymbol{x}<\boldsymbol{l}, \boldsymbol{t}>0 \\\left.\boldsymbol{u}\right|_{\boldsymbol{x}=0}=\left.\boldsymbol{u}\right|_{\boldsymbol{x}=l}=0 \\\left.\boldsymbol{u}\right|_{t=0}=\phi(\boldsymbol{x}),\left.\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{t}}\right|_{t=0}=\varPsi(\boldsymbol{x})\end{array}\right. ⎩ ⎨ ⎧∂ t2∂2 and=a2∂ x2∂2 and+f(x,t),0<x<l,t>0u ∣x=0​=u ∣x=l​=0u ∣t=0​=ϕ ( x ) ,∂ t∂ u∣ ∣t=0​=Ψ ( x )
拆解$u(x,t)=v(x,t)+w(x,t)$

2.4.1 Eigenfunction method

wave equation

{ ∂ 2 v ∂ t 2 = a 2 ∂ 2 v ∂ x 2 + f ( x , t ) v ∣ x = 0 = v ∣ x = l = 0 v ∣ t = 0 = 0 , ∂ v ∂ t ∣ t = 0 = 0 \left\{\begin{array}{l}\frac{\partial^{2} v}{\partial t^{2}}=a^{2} \frac{\partial^{2} v}{\partial x^{2}}+f(x, t) \\\left.v\right|_{x=0}=\left.v\right|_{x=l}=0 \\\left.v\right|_{t=0}=0,\left.\frac{\partial v}{\partial t}\right|_{t=0}=0\end{array}\right. ⎩ ⎨ ⎧∂ t2∂2 v=a2∂ x2∂2 v+f(x,t)v ∣x=0​=v ∣x=l​=0v ∣t=0​=0 ,∂ t∂ v∣ ∣t=0​=0

1. Homogeneous equation + homogeneous boundary condition $\{\begin{array}{l}\frac{{\partial }^2}{\partial t^2}={\mathbf{a}}^2\frac{{\partial }^2}{\partial x^2}\\ {v|}_{\mathbf{x}=0}={v|}_{\mathbf{x}=l}=\end{array}$
   1. Eigenvalue problem $\{\begin{array}{l}{\mathbf{X}}^{\prime \prime }(\mathbf{x})+\lambda \mathbf{X}(\mathbf{x})=0\\ \mathbf{X}(0)=\mathbf{X}(\mathbf{l})=\end{array}$
      1. Characteristic function $X_n(x)=B_n\sin \frac{n\pi }{l}x,n=1,2,...$
      2. Suppose the inhomogeneous equation solution $v(x,t)={\sum }_{n=1}^{}{in}_n(t)\sin \frac{n\pi }{l}x$
2. $f(x,t)$ according to the sequence ofeigenfunctions { sin ⁡ n π lx } n = 1 ∞ \{\sin\frac{n\pi}{l}x\}_{n=1}^{\infty}{ sinlnπ​x}n=1∞Expand into series form
   1. $f(x,t)={\sum }_{n=1}^{}{f}_n\left(t\right)\sin \frac{n\pi }{l}x$
   2. $f_n\left(t\right)=\frac{}{l}{\int }_0^{l}f(x,t)\sin \frac{n\pi }{l}xdx$
3. The above results are brought into the inhomogeneous equation
   1. 
   2. Get the ordinary differential equation problem $\{\begin{array}{l}{in}_n^{\prime }(t)+{\left(\frac{}{l}\right)}^2{in}_n\left(t\right)=f_n\left(t\right)\\ {in}_n\left(0\right)=0,{in}_n^{}\left(0\right)=\end{array}$
   3. Laplace Transform
      1. ${in}_n(t)=\frac{l}{n\pi a}{\int }_0^t{f}_n\left(t\right)\sin \frac{npa(t-\tau }{l}d\tau$
      2. $\mathbf{v}(\mathbf{x},\mathbf{t})={\sum }_{n=1}^{}\frac{l}{n\pi a}{\int }_0^t{f}_n\left(t\right)\sin \frac{npa(t-\tau }{l}d\tau \sin \frac{n\pi }{l}x$
         Matlab Solving Second Order Ordinary Differential Equations

syms V n a L 
S=dsolve(`D2V+(n*pi*a/L)^2*V=5`,`V(0)=0,DV(0)=0`,`t`)
pretty(simple(S))

heat equation

{ ∂ u ∂ t = a 2 ∂ 2 u ∂ x 2 + sin ⁡ ω t 0 < x < l , t > 0 ∂ u ∂ x ∣ x = 0 = ∂ u ∂ x ∣ x = l = 0 u ∣ t = 0 = 0 \left\{\begin{array}{l} \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{t}}=\boldsymbol{a}^{2} \frac{\partial^{2} \boldsymbol{u}}{\partial \boldsymbol{x}^{2}}+\sin \omega \boldsymbol{t} \quad 0<\boldsymbol{x}<\boldsymbol{l}, \boldsymbol{t}>0 \\ \left.\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}}\right|_{x=0}=\left.\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}}\right|_{x=l}=0 \\ \left.\boldsymbol{u}\right|_{t=0}=0 \end{array}\right. ⎩ ⎨ ⎧∂ t∂ u=a2∂ x2∂2 and+sint _0<x<l,t>0∂ x∂ u∣ ∣x=0​=∂ x∂ u∣ ∣x=l​=0u ∣t=0​=0

1. 齐次方程+齐次边界条件$\{\begin{array}{l}\frac{\partial }{\partial t}={\mathbf{a}}^2\frac{{\partial }^2}{\partial x^2}\\ {\frac{\partial }{\partial x}|}_{x=0}={\frac{\partial }{\partial x}|}_{x=l}=\end{array}$
   1. Eigenvalue problem $\{\begin{array}{l}{\mathbf{X}}^{\prime \prime }(\mathbf{x})+\lambda \mathbf{X}(\mathbf{x})=0\\ X^{\prime }\left(0\right)=X^{\prime }(\mathbf{l})=\end{array}$
   2. Characteristic function $X_n(x)=A_n\cos \frac{n\pi }{l}x,n=0,1,2,...$
   3. Let inhomogeneous equation solution $u(x,t)={\sum }_{n=1}^{}{in}_n(t)\cos \frac{n\pi }{l}x$
2. $\sin wt$ according to the sequence ofeigenfunctions { cos ⁡ n π lx } n = 0 ∞ \{\cos\frac{n\pi}{l}x\}_{n=0}^\infty{ coslnπ​x}n=0∞Expand into series form
   1. $\sin wt=f_{}+{\sum }_{n=1}^{}{f}_n\left(t\right)\cos \frac{n\pi }{l}x$
      1. $f_{}\left(t\right)=\frac{}{l}{\int }_0^l\sin wtdx=\sin wt$
      2. $f_n(t)=\frac{}{l}{\int }_0^l\sin wt\cos \frac{n\pi }{l}xdx=0$
3. Substitute into the inhomogeneous equation to get
   1. 
   2. 得常微分方程问题$\{\begin{array}{l}{in}_n^{}\left(t\right)+{\left(\frac{}{l}\right)}^2{in}_n\left(t\right)=f_n\left(t\right)\\ {in}_n\left(0\right)=\end{array}$
      1. $n=0$得$\{\begin{array}{l}{in}_0^{}(\mathbf{t})=\sin \mathbf{w}\mathbf{t}\\ {in}_{}\left(0\right)=\end{array}$
         1. $\begin{array}{c}{in}_{}\left(t\right)=-\frac{}{in}\cos wt+C\\ {in}_{}\left(0\right)=\end{array}\}\Rightarrow {in}_{}\left(t\right)=\frac{}{in}(1-\cos wt)$
      2. $n\ne 0$得$\{\begin{array}{l}{in}_n^{}(t)+{\left(\frac{}{l}\right)}^2{in}_n\left(t\right)=0\\ {in}_n\left(0\right)=\end{array}$
         1. $\begin{array}{l}{in}_n\left(t\right)=This{\_}^{-a^2\frac{n^2{p.m}^{}}{l^2}t}\\ {in}_n\left(0\right)=\end{array}\}\Rightarrow {in}_n\left(t\right)\equiv 0$
   3. $u(x,t)={\sum }_{n=0}^{}{in}_n(t)\cos \frac{n\pi }{l}x$

2.5 Treatment of inhomogeneous boundary conditions

2.4波方程$u(x,t)=v(x,t)+w(x,t)$

1. v at the boundary satisfies $v(o,t)=0,v(l,t)=0$ then w satisfies$in(0,t)={in}_{}(t),w(l,t)={in}_{}(t)$
2. w形式$w(x,t)=A(t)x+B(t)$
   1. 满足$\{\begin{array}{l}w(x,t)=A\left(t\right)x+B\left(t\right)\\ in(0,t)={in}_{}\left(t\right),w(l,t)={in}_{}(t)\end{array}$
   2. 解得$\{\begin{array}{l}A(t)=\frac{{in}_{}(t)-u_{}(t)}{l}\\ B(t)={in}_{}\left(t\right)\end{array}$
   3. $\therefore w(x,t)=\frac{{in}_{}\left(t\right)-u_{}\left(t\right)}{l}x+{in}_{}\left(t\right)$
3. in
   1. $v(x,t)$满足 { ∂ 2 v ∂ t 2 = a 2 ∂ 2 v ∂ x 2 + f 1 ( x , t ) v ∣ x = 0 = 0 , v ∣ x = l = 0 v ∣ t = 0 = ϕ 1 ( x ) , ∂ v ∂ t ∣ t = 0 = ψ 1 ( x ) \left\{\begin{array}{l}\frac{\partial^{2} \boldsymbol{v}}{\partial \boldsymbol{t}^{2}}=\boldsymbol{a}^{2} \frac{\partial^{2} \boldsymbol{v}}{\partial \boldsymbol{x}^{2}}+\boldsymbol{f}_{1}(\boldsymbol{x}, \boldsymbol{t}) \\\left.\boldsymbol{v}\right|_{x=0}=0,\left.\quad \boldsymbol{v}\right|_{x=l}=0 \\\left.\boldsymbol{v}\right|_{t=0}=\phi_{1}(\boldsymbol{x}),\left. \frac{\partial \boldsymbol{v}}{\partial \boldsymbol{t}}\right|_{t=0}=\psi_{1}(\boldsymbol{x})\end{array}\right. ⎩ ⎨ ⎧∂ t2∂2 v=a2∂ x2∂2 v+f1(x,t)v ∣x=0​=0 ,v ∣x=l​=0v ∣t=0​=ϕ1(x),∂ t∂ v∣ ∣t=0​=p1(x)​in$\begin{array}{l}{f}_{}(x,t)=f(x,t)-\frac{{in}_2^{\prime }(\mathbf{t})-{\mathbf{u}}_1^{\prime }(\mathbf{t})}{l}x-{in}_1^{\prime }\left(t\right)\\ {\varphi }_{}\left(x\right)=\varphi \left(x\right)-{in}_{}\left(0\right)-\frac{{in}_{}\left(0\right)-u_{}(0}{l}x\\ p_{}\left(x\right)=\psi \left(x\right)-{in}_1^{}\left(0\right)-\frac{{in}_2^{}\left(0\right)-u_1^{}(0}{l}x\end{array}$