Partial Differential Equations

UPSC Mathematics Optional · Paper II · Classification to Fourier Methods

1. Formation & Classification 2. First Order PDEs — Lagrange's Method 3. Charpit's & Jacobi's Methods 4. Second Order — Classification 5. Wave Equation 6. Heat Equation 7. Laplace Equation 8. Fourier Series 9. Fourier Transform

§1 · Formation & Order

Notation

\(p=\dfrac{\partial z}{\partial x}\), \(q=\dfrac{\partial z}{\partial y}\), \(r=\dfrac{\partial^2 z}{\partial x^2}\), \(s=\dfrac{\partial^2 z}{\partial x\partial y}\), \(t=\dfrac{\partial^2 z}{\partial y^2}\).

Formation: eliminate arbitrary constants/functions by differentiation.

§2 · First Order PDEs — Lagrange's Method

Lagrange's Linear PDE PYQ★★★

\(Pp+Qq=R\) where \(P,Q,R\) are functions of \(x,y,z\).

Lagrange's auxiliary (subsidiary) equations: \(\dfrac{dx}{P}=\dfrac{dy}{Q}=\dfrac{dz}{R}\)

Solve for two independent integrals \(u(x,y,z)=a\) and \(v(x,y,z)=b\). General solution: \(\phi(u,v)=0\) or \(u=f(v)\).

Tricks for auxiliary equations: use multipliers \((l,m,n)\) s.t. \(lP+mQ+nR=0\) → \(l\,dx+m\,dy+n\,dz=0\).

§3 · Charpit's Method & Non-linear First Order PDEs

Charpit's Method PYQ★★★

For \(F(x,y,z,p,q)=0\). Charpit's auxiliary equations:

\[\frac{dp}{F_x+pF_z}=\frac{dq}{F_y+qF_z}=\frac{dz}{-pF_p-qF_q}=\frac{dx}{-F_p}=\frac{dy}{-F_q}\]

Find one integral involving \(p\) or \(q\) → substitute back → integrate for complete integral.

Special Types of First Order PDEs PYQ★★★

Type I: \(F(p,q)=0\) → assume \(z=ax+by+c\) with \(F(a,b)=0\).
Type II: \(F(z,p,q)=0\) → assume \(z=f(ax+by)\) → ODE for \(f\).
Type III (Clairaut): \(z=px+qy+f(p,q)\) → complete integral: replace \(p\to a\), \(q\to b\).
Type IV: Separable: \(F(x,p)=G(y,q)\) → set both = constant \(a\) → integrate.

§4 · Second Order PDEs — Classification

Classification of \(Ar+Bs+Ct=F(x,y,z,p,q)\) PYQ★★★

Discriminant \(\Delta=B^2-4AC\):

\(\Delta>0\): Hyperbolic (e.g. wave equation). Two real characteristic families.
\(\Delta=0\): Parabolic (e.g. heat equation). One repeated characteristic family.
\(\Delta<0\): Elliptic (e.g. Laplace equation). No real characteristics.

Canonical forms obtained by transforming to characteristic coordinates.

Second Order PDEs with Constant Coefficients PYQ★★★

CF of \(f(D,D')z=0\) where \(D=\partial/\partial x\), \(D'=\partial/\partial y\).

Auxiliary: \(f(m,1)=0\) gives \(m_1,m_2,\ldots\) (put \(D\to m\), \(D'\to 1\)).

Distinct \(m_i\): CF \(=\phi_1(y+m_1 x)+\phi_2(y+m_2 x)+\cdots\)
Repeated root \(m\) (twice): CF \(=\phi_1(y+mx)+x\phi_2(y+mx)\)

PI: \(\dfrac{1}{f(D,D')}e^{ax+by}=\dfrac{e^{ax+by}}{f(a,b)}\) (if \(f(a,b)\ne 0\)).

\(\dfrac{1}{f(D,D')}\sin(ax+by)=\dfrac{\sin(ax+by)}{f(-a^2,-ab,-b^2)}\) (replace \(D^2\to-a^2\), \(D D'\to-ab\), \(D'^2\to-b^2\)).

§5 · Wave Equation

1D Wave Equation \(\dfrac{\partial^2 u}{\partial t^2}=c^2\dfrac{\partial^2 u}{\partial x^2}\) PYQ★★★

D'Alembert's solution: \(u(x,t)=f(x+ct)+g(x-ct)\).

With \(u(x,0)=\phi(x)\), \(u_t(x,0)=\psi(x)\):

\[u(x,t)=\frac{\phi(x+ct)+\phi(x-ct)}{2}+\frac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds\]

Separation of variables (finite string, \(0\le x\le L\)):

\(u(x,t)=\sum_{n=1}^\infty\sin\dfrac{n\pi x}{L}\left(A_n\cos\dfrac{n\pi ct}{L}+B_n\sin\dfrac{n\pi ct}{L}\right)\)

where \(A_n,B_n\) from Fourier series of \(\phi,\psi\).

§6 · Heat Equation

1D Heat Equation \(\dfrac{\partial u}{\partial t}=k\dfrac{\partial^2 u}{\partial x^2}\) PYQ★★★

Separation of variables: \(u(x,t)=X(x)T(t)\) → \(X''/X=T'/(kT)=-\lambda^2\).

With \(u(0,t)=u(L,t)=0\), \(u(x,0)=f(x)\):

\[u(x,t)=\sum_{n=1}^\infty B_n\sin\frac{n\pi x}{L}\cdot e^{-k(n\pi/L)^2 t}\]

\(B_n=\dfrac{2}{L}\int_0^L f(x)\sin\dfrac{n\pi x}{L}\,dx\).

Steady state: As \(t\to\infty\), transient terms die out; only constant remains if non-zero BCs.

§7 · Laplace Equation

Laplace Equation \(\nabla^2 u=0\) PYQ★★★

Elliptic. Solutions = harmonic functions.

Rectangular domain: \(u(x,y)=\sum(A_n\sinh\dfrac{n\pi y}{L}+B_n\cosh\dfrac{n\pi y}{L})\sin\dfrac{n\pi x}{L}+\cdots\)

Polar coordinates: \(u=\sum(A_n r^n+B_n r^{-n})(C_n\cos n\theta+D_n\sin n\theta)\).

For disk \(r

\[u(r,\theta)=\frac{1}{2\pi}\int_0^{2\pi}\frac{a^2-r^2}{a^2-2ar\cos(\theta-\phi)+r^2}f(\phi)\,d\phi\]

Maximum principle: Harmonic \(u\) attains max and min on boundary.

§8 · Fourier Series

Fourier Series PYQ★★★

\[f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty\left(a_n\cos\frac{n\pi x}{L}+b_n\sin\frac{n\pi x}{L}\right)\]

\[a_0=\frac{1}{L}\int_{-L}^L f(x)\,dx,\quad a_n=\frac{1}{L}\int_{-L}^L f(x)\cos\frac{n\pi x}{L}\,dx,\quad b_n=\frac{1}{L}\int_{-L}^L f(x)\sin\frac{n\pi x}{L}\,dx\]

Dirichlet conditions: \(f\) piecewise smooth → Fourier series converges to \(f(x)\) at continuity points, to \(\frac{f(x^+)+f(x^-)}{2}\) at jumps.

Half-range expansions: Even → cosine series; Odd → sine series.

Parseval's identity: \(\dfrac{1}{L}\int_{-L}^L[f(x)]^2\,dx=\dfrac{a_0^2}{2}+\sum_{n=1}^\infty(a_n^2+b_n^2)\)

§9 · Fourier Transform

Fourier Transform Pair PYQ★★

\[\hat f(\xi)=\int_{-\infty}^\infty f(x)e^{-2\pi i\xi x}\,dx, \qquad f(x)=\int_{-\infty}^\infty\hat f(\xi)e^{2\pi i\xi x}\,d\xi\]

(Convention varies: UPSC typically uses \(e^{i\omega x}\) form with \(\frac{1}{2\pi}\) factor.)

Key properties: Linearity, shifting, scaling, differentiation (\(\mathcal{F}[f']=i\omega\mathcal{F}[f]\)), convolution (\(\mathcal{F}[f*g]=\mathcal{F}[f]\cdot\mathcal{F}[g]\)).

Parseval: \(\int|f|^2\,dx=\int|\hat f|^2\,d\xi\).

Fourier sine/cosine transforms: used when \(f\) defined on \([0,\infty)\).