Laplace on rectangle — separation of variables

4.7.14 · Maths › Partial Differential Equations

YEH problem KYA hai?

Ek waqt mein sirf ek nonzero side KYU? Laplace's equation linear aur homogeneous hai, isliye solutions ke sums bhi solutions hote hain. Agar chaaron sides nonzero hain, toh chaar sub-problems mein toddo jisme har ek mein teen zero edges hain, har ek solve karo, aur jod do. Toh ek-side case master karna matlab sab kuch master karna.

KAISE: scratch se derive karo

Step 1 — Separate karo

Maano $u(x,y)=X(x)Y(y)$. Plug in karo: $X^{\prime \prime }(x)Y(y)+X(x)Y^{\prime \prime }(y)=0.$ $X(x)Y(y)$ se divide karo (valid hai jab dono nonzero ho): $\frac{X^{\prime \prime }}{X}+\frac{Y^{\prime \prime }}{Y}=0⟹\frac{X^{\prime \prime }}{X}=-\frac{Y^{\prime \prime }}{Y}.$

$\frac{X^{\prime \prime }}{X}=-\lambda ,\frac{Y^{\prime \prime }}{Y}=\lambda .$

Step 2 — Homogeneous direction use karke sign choose karo

Hamare paas $u(0,y)=u(a,y)=0$ hai, yaani $X(0)=X(a)=0$. $x$ mein do zero boundary conditions ⇒ hume ek oscillating $X$ chahiye (ek sine), jo $\lambda >0$ force karta hai. $\lambda =k^2$ likho.

Step 3 — $X$ eigenvalue problem solve karo

$X^{\prime \prime }+k^{2}X=0\Rightarrow X=A\cos kx+B\sin kx.$ $X(0)=0\Rightarrow A=0$. Toh $X=B\sin kx$. $X(a)=0\Rightarrow \sin ka=0\Rightarrow ka=n\pi$, $n=1,2,3,\dots$

$\boxed{k_n=\frac{n\pi }{a},X_n(x)=\sin \frac{n\pi x}{a}}$

Yeh eigenfunctions hain; ${\lambda }_n=(n\pi /a{)}^2$ eigenvalues hain.

Step 4 — Matching $\lambda$ ke saath $Y$ solve karo

$Y^{\prime \prime }-k_n^{2}Y=0\Rightarrow Y=C\cosh \frac{n\pi y}{a}+D\sinh \frac{n\pi y}{a}.$ $u(x,0)=0\Rightarrow Y(0)=0\Rightarrow C=0$ apply karo. Toh $Y_n(y)=\sinh \frac{n\pi y}{a}.$

Step 5 — Superpose karo aur last edge match karo

Har $u_n=\sin \frac{n\pi x}{a}\sinh \frac{n\pi y}{a}$ teen zero edges ke saath Laplace solve karta hai. Inhe sum karo: $u(x,y)={\sum }_{n=1}^{\infty }{B}_n\sin \frac{n\pi x}{a}\sinh \frac{n\pi y}{a}.$ Final condition $u(x,b)=f(x)$ impose karo: $f(x)={\sum }_{n=1}^{\infty }\left(B_n\sinh \frac{n\pi b}{a}\right)\sin \frac{n\pi x}{a}.$ Yeh sirf $f$ ki Fourier sine series hai. Bracket coefficient hai: $B_n\sinh \frac{n\pi b}{a}=\frac{2}{a}{\int }_0^{a}f(x)\sin \frac{n\pi x}{a}dx.$

Worked examples

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Connections

• Fourier sine series — woh engine jo $B_n$ compute karta hai.
• Separation of variables — general method (bhi use hota hai Heat equation 1D, Wave equation 1D ke liye).
• Sturm–Liouville eigenvalue problems — kyun $\sin (n\pi x/a)$ orthogonal eigenfunctions hain.
• Hyperbolic functions sinh cosh — decaying/growing $y$-profiles.
• Superposition principle — chaar-edge problem ko split karna.
• Harmonic functions and mean value property — "neighbours ki average" wali picture.
• Heat equation 1D — same eigenfunctions, lekin $\sinh$ ki jagah $e^{-\lambda t}$.

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