CSIR NET Mathematical Science UNIT – 3 Syllabus

List the defining properties of the Green's function for a second order Sturm–Liouville operator.

1) $G$ satisfies the homogeneous ODE for $x\neq\xi$. 2) $G$ satisfies the boundary conditions. 3) $G$ is continuous at $x=\xi$. 4) Its derivative has a jump: $\frac{\partial G}{\partial x}\big|_{\xi^+} - \frac{\partial G}{\partial x}\big|_{\xi^-} = \frac{1}{p(\xi)}$. Also $G$ is symmetric: $G(x,\xi)=G(\xi,x)$.

How is Green's function constructed from two solutions $y_1,y_2$ satisfying the left and right boundary conditions?

$$G(x,\xi)=\begin{cases} \dfrac{y_1(x)\,y_2(\xi)}{p(\xi)\,W(\xi)}, & x\leq \xi \\[2mm] \dfrac{y_1(\xi)\,y_2(x)}{p(\xi)\,W(\xi)}, & x\geq \xi \end{cases}$$ where $y_1$ meets the left BC, $y_2$ the right BC, and $W$ is their Wronskian.

State Lagrange's method for solving the first order linear (quasilinear) PDE $Pp + Qq = R$ where $p=z_x,\,q=z_y$.

Form Lagrange's auxiliary (subsidiary) equations $$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}.$$ Find two independent integrals $u=c_1,\,v=c_2$. The general solution is $\Phi(u,v)=0$ (or $u=\phi(v)$).

Outline Charpit's method for the general first order nonlinear PDE $F(x,y,z,p,q)=0$.

Write Charpit's auxiliary equations: $$\frac{dx}{F_p}=\frac{dy}{F_q}=\frac{dz}{pF_p+qF_q}=\frac{dp}{-(F_x+pF_z)}=\frac{dq}{-(F_y+qF_z)}.$$ Find a relation $f(x,y,z,p,q,a)=0$, solve with $F=0$ for $p,q$, then integrate $dz=p\,dx+q\,dy$.

Write the four standard types of first order nonlinear PDEs handled by special methods (before Charpit).

Type I: $f(p,q)=0$ (only $p,q$). Type II: $z=px+qy+f(p,q)$ (Clairaut form). Type III: $f(z,p,q)=0$ ($x,y$ absent). Type IV: $f_1(x,p)=f_2(y,q)$ (separable). Each has a complete integral $z=ax+by+c$ style ansatz.

What is the Cauchy problem for a first order PDE, and what condition ensures a unique solution near the initial curve?

Find a solution surface $z=z(x,y)$ containing a prescribed initial curve $\Gamma:(x_0(s),y_0(s),z_0(s))$. Uniqueness holds when the transversality (non-characteristic) condition is met: the initial curve is nowhere tangent to the characteristic directions, i.e. $\begin{vmatrix} x_0'(s) & y_0'(s) \\ P & Q\end{vmatrix}\neq 0$.

Give the classification of the general second order linear PDE $A u_{xx}+B u_{xy}+C u_{yy}+\dots=0$.

Based on discriminant $B^2-4AC$: \n- $B^2-4AC>0$: hyperbolic (e.g. wave equation). \n- $B^2-4AC=0$: parabolic (e.g. heat equation). \n- $B^2-4AC<0$: elliptic (e.g. Laplace equation).

Classify the wave, heat, and Laplace equations and give their canonical forms.

Wave $u_{tt}=c^2u_{xx}$: hyperbolic. Heat $u_t=\alpha u_{xx}$: parabolic. Laplace $u_{xx}+u_{yy}=0$: elliptic. They are the prototypes of the three PDE classes.

How do you find the general solution of a higher order linear PDE with constant coefficients $F(D,D')z=0$ where $D=\partial_x,\,D'=\partial_y$?

If $F(D,D')$ factors into linear factors $(D-m_iD')$, each simple factor $(D-mD')$ contributes $\phi_i(y+mx)$. A factor repeated $k$ times contributes $\phi_1(y+mx)+x\phi_2(y+mx)+\dots+x^{k-1}\phi_k(y+mx)$, with arbitrary functions $\phi_i$.

For $F(D,D')z=0$, how are the roots $m$ found, and what is the complementary function for distinct roots $m_1,\dots,m_n$?

Put $D=m,\,D'=1$ in the homogeneous operator to get the auxiliary equation $F(m,1)=0$. For distinct roots: $$z = \phi_1(y+m_1x)+\phi_2(y+m_2x)+\cdots+\phi_n(y+m_n x).$$

Apply separation of variables to the 1-D heat equation $u_t=\alpha u_{xx}$. What ODEs and solution form arise?

Let $u=X(x)T(t)$. Then $\frac{T'}{\alpha T}=\frac{X''}{X}=-\lambda$. So $X''+\lambda X=0$ and $T'+\alpha\lambda T=0$. For $\lambda=k^2>0$: $T=e^{-\alpha k^2 t}$, giving solutions $u=e^{-\alpha k^2 t}(A\cos kx+B\sin kx)$.

Apply separation of variables to the 1-D wave equation $u_{tt}=c^2u_{xx}$. What ODEs result?

Let $u=X(x)T(t)$. Then $\frac{T''}{c^2T}=\frac{X''}{X}=-\lambda$, giving $X''+\lambda X=0$ and $T''+c^2\lambda T=0$. Both factors are oscillatory for $\lambda>0$, e.g. $u=(A\cos kx+B\sin kx)(C\cos ckt+D\sin ckt)$ with $\lambda=k^2$.

Apply separation of variables to Laplace's equation $u_{xx}+u_{yy}=0$ in Cartesian coordinates.

Let $u=X(x)Y(y)$. Then $\frac{X''}{X}=-\frac{Y''}{Y}=-\lambda$. So $X''+\lambda X=0$ (oscillatory) and $Y''-\lambda Y=0$ (exponential/hyperbolic) for $\lambda>0$, e.g. $u=(A\cos kx+B\sin kx)(Ce^{ky}+De^{-ky})$.

State the bisection method and its convergence behaviour for finding a root of $f(x)=0$.

If $f$ is continuous on $[a,b]$ with $f(a)f(b)<0$, repeatedly bisect: $c=\frac{a+b}{2}$, keep the subinterval where the sign change occurs. It always converges (linearly) with error halving each step: after $n$ steps error $\leq \frac{b-a}{2^{n}}$.

Write the Newton–Raphson iteration formula and state its order of convergence.

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.$$ It converges quadratically (order $2$) to a simple root, provided $f'\neq 0$ near the root and the initial guess is sufficiently close.

Define the fixed-point iteration method and state its convergence condition.

Rewrite $f(x)=0$ as $x=g(x)$ and iterate $x_{n+1}=g(x_n)$. It converges to a fixed point $\alpha$ (linearly) if $g$ is continuous and $|g'(x)|<1$ in a neighbourhood of $\alpha$; smaller $|g'(\alpha)|$ gives faster convergence.

Define the order (rate) of convergence $p$ of an iterative method with errors $e_n=x_n-\alpha$.

The method has order $p$ if $$\lim_{n\to\infty}\frac{|e_{n+1}|}{|e_n|^{p}} = C \neq 0$$ for some constant $C$ (the asymptotic error constant). $p=1$ is linear, $p=2$ quadratic, $p\approx1.618$ for the secant method.

Give the secant method formula and its order of convergence.

$$x_{n+1} = x_n - f(x_n)\,\frac{x_n - x_{n-1}}{f(x_n)-f(x_{n-1})}.$$ Order of convergence is the golden ratio $p=\frac{1+\sqrt{5}}{2}\approx 1.618$ (superlinear).

What is the order of convergence of Newton–Raphson at a root of multiplicity $m>1$, and how can quadratic convergence be restored?

At a multiple root, ordinary Newton–Raphson converges only linearly (rate $1-\tfrac{1}{m}$). Modified Newton $x_{n+1}=x_n - m\frac{f(x_n)}{f'(x_n)}$ restores quadratic convergence.

Describe the Gauss elimination method for solving $A\vec{x}=\vec{b}$.

Use elementary row operations (forward elimination) to reduce the augmented matrix $[A\,|\,b]$ to upper-triangular form, then solve by back substitution. Partial pivoting (swapping rows to put the largest pivot on the diagonal) is used to control round-off error. Operation count $\sim \frac{n^3}{3}$.

Write the Gauss–Seidel iteration formula for the system $A\vec{x}=\vec{b}$.

$$x_i^{(k+1)} = \frac{1}{a_{ii}}\left(b_i - \sum_{j<i} a_{ij}x_j^{(k+1)} - \sum_{j>i} a_{ij}x_j^{(k)}\right).$$ It uses the most recently updated values immediately within each sweep.

How does Gauss–Seidel differ from the Jacobi method, and what guarantees convergence?

Jacobi uses only old iterate values $x_j^{(k)}$ for all $j$; Gauss–Seidel uses updated values $x_j^{(k+1)}$ for $j<i$ within the same sweep, so it generally converges faster. Both converge if $A$ is strictly diagonally dominant (or symmetric positive definite for Gauss–Seidel).

Define the forward, backward, and central difference operators.

Forward: $\Delta f(x)=f(x+h)-f(x)$. Backward: $\nabla f(x)=f(x)-f(x-h)$. Central: $\delta f(x)=f\!\left(x+\tfrac{h}{2}\right)-f\!\left(x-\tfrac{h}{2}\right)$.

State the relations between the shift operator $E$ and the forward/backward difference operators.

$Ef(x)=f(x+h)$. Then $\Delta = E-1$, $\nabla = 1-E^{-1}$, and $E=e^{hD}$ where $D$ is the differentiation operator. Also $\delta = E^{1/2}-E^{-1/2}$ and the averaging operator $\mu=\tfrac{1}{2}(E^{1/2}+E^{-1/2})$.

Write the Newton's forward difference interpolation formula.

With $u=\frac{x-x_0}{h}$: $$f(x)=f_0 + u\Delta f_0 + \frac{u(u-1)}{2!}\Delta^2 f_0 + \frac{u(u-1)(u-2)}{3!}\Delta^3 f_0 + \cdots$$

State the Lagrange interpolation formula for nodes $x_0,\dots,x_n$.

$$P(x)=\sum_{i=0}^{n} f(x_i)\,L_i(x),\qquad L_i(x)=\prod_{\substack{j=0\\ j\neq i}}^{n}\frac{x-x_j}{x_i-x_j}.$$ The $L_i$ satisfy $L_i(x_j)=\delta_{ij}$.

How does Hermite interpolation differ from Lagrange interpolation, and what is the degree of the Hermite polynomial through $n+1$ nodes?

Hermite interpolation matches both function values $f(x_i)$ and derivative values $f'(x_i)$ at each node. With $n+1$ nodes it imposes $2(n+1)$ conditions, giving a unique polynomial of degree $\leq 2n+1$.

What is a cubic spline interpolant and what continuity/smoothness does it enforce?

A cubic spline is a piecewise cubic polynomial on each subinterval $[x_i,x_{i+1}]$ that interpolates the data and is $C^2$ globally: the function, first derivative, and second derivative are continuous across all interior nodes. A natural spline additionally sets $S''=0$ at the endpoints.

State the error/remainder term for polynomial interpolation at $n+1$ nodes.

$$f(x)-P_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^{n}(x-x_i),$$ for some $\xi$ in the smallest interval containing the nodes and $x$ (assuming $f\in C^{n+1}$).

Compare direct (Gauss elimination) and iterative (Gauss–Seidel) methods for linear systems.

Direct methods (Gauss elimination) give the exact solution in finitely many steps ($\sim n^3/3$ operations), best for small/dense systems. Iterative methods (Gauss–Seidel/Jacobi) refine an approximation each sweep, are memory-efficient for large sparse systems, but require a convergence condition (e.g. diagonal dominance).