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GATE Mathematics Partial Differential Equations Syllabus
The Partial Differential Equations syllabus for GATE Mathematics in full: 4 chapters and 6 topics, every one listed below, plus 51 Partial Differential Equations flashcards with answers.
Partial Differential Equations syllabus: all 4 chapters
Expand any chapter to see its topics and sub-topics.
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Method of Characteristics for First Order Linear and Quasilinear Partial Differential Equations
1 topic- Method of Characteristics
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Second Order Partial Differential Equations in Two Independent Variables: Classification and Canonical Forms
1 topic- Classification and Canonical Forms
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Method of Separation of Variables for Laplace Equation in Cartesian and Polar Coordinates
2 topics- Laplace Equation in Cartesian Coordinates
- Laplace Equation in Polar Coordinates
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Heat and Wave Equations in One Space Variable
2 topics- Heat Equation
- Wave Equation
GATE Mathematics Partial Differential Equations flashcards — 51 questions with answers
What is the general first-order quasilinear PDE solved by the method of characteristics?
A PDE of the form $a(x,y,u)\,u_x + b(x,y,u)\,u_y = c(x,y,u)$, where the solution surface is built from characteristic curves.
Write the characteristic (Lagrange–Charpit) equations for $a\,u_x + b\,u_y = c$.
$$\frac{dx}{a} = \frac{dy}{b} = \frac{du}{c}$$ These ODEs are integrated to obtain two independent first integrals.
Using Lagrange's method, how is the general solution of $a u_x + b u_y = c$ expressed?
If the characteristic equations give independent integrals $\phi(x,y,u)=c_1$ and $\psi(x,y,u)=c_2$, the general solution is $F(\phi,\psi)=0$ for an arbitrary function $F$.
Solve $u_x + u_y = 0$ by characteristics. What is the general solution?
The characteristics are $\frac{dx}{1}=\frac{dy}{1}=\frac{du}{0}$, giving $x-y=c_1$ and $u=c_2$. Thus $u(x,y)=f(x-y)$ for arbitrary $f$.
Along a characteristic curve of a first-order PDE, how does $u$ behave for a homogeneous equation ($c=0$)?
Since $\frac{du}{c}$ with $c=0$ forces $du=0$, the solution $u$ is constant along each characteristic curve.
For the transport equation $u_t + c\,u_x = 0$ with $u(x,0)=f(x)$, what is the solution?
$u(x,t)=f(x-ct)$ — the initial profile travels with speed $c$ unchanged.
What is the general form of a second-order linear PDE in two variables?
$$A u_{xx} + B u_{xy} + C u_{yy} + D u_x + E u_y + F u = G$$ where $A,B,C,\dots$ may depend on $x,y$.
What discriminant classifies a second-order PDE $A u_{xx}+B u_{xy}+C u_{yy}+\dots=0$?
The discriminant $B^{2}-4AC$ determines the type at each point.
State the classification of a second-order PDE based on $B^{2}-4AC$.
Hyperbolic if $B^{2}-4AC>0$; Parabolic if $B^{2}-4AC=0$; Elliptic if $B^{2}-4AC<0$.
Classify the wave equation $u_{tt}=c^{2}u_{xx}$.
Here $A=c^{2},B=0,C=-1$ (in $x,t$), so $B^{2}-4AC=4c^{2}>0$: it is hyperbolic.
Classify the heat equation $u_t = k\,u_{xx}$.
With second-order terms only $u_{xx}$ ($A=k,B=0,C=0$), $B^{2}-4AC=0$: it is parabolic.
Classify the Laplace equation $u_{xx}+u_{yy}=0$.
$A=1,B=0,C=1$, so $B^{2}-4AC=-4<0$: it is elliptic.
What is the canonical form of a hyperbolic equation?
Using characteristic coordinates $\xi,\eta$, the canonical form is $u_{\xi\eta}=\Phi(\xi,\eta,u,u_\xi,u_\eta)$ (or equivalently $u_{\alpha\alpha}-u_{\beta\beta}=\dots$).
What is the canonical form of a parabolic equation?
It reduces to $u_{\eta\eta}=\Phi(\xi,\eta,u,u_\xi,u_\eta)$, with only one second-derivative term.
What is the canonical form of an elliptic equation?
Using $\alpha,\beta$ coordinates it becomes $u_{\alpha\alpha}+u_{\beta\beta}=\Phi(\alpha,\beta,u,u_\alpha,u_\beta)$.
What ODE gives the characteristic curves of $A u_{xx}+B u_{xy}+C u_{yy}=0$?
The characteristic equation $A\,(dy)^{2} - B\,dx\,dy + C\,(dx)^{2}=0$, i.e. $A\left(\frac{dy}{dx}\right)^{2}-B\frac{dy}{dx}+C=0$.
How many real characteristic families do hyperbolic, parabolic and elliptic equations have?
Hyperbolic: two real distinct families; Parabolic: one real (repeated) family; Elliptic: no real characteristics (complex).
What is the Laplace equation in two-dimensional Cartesian coordinates?
$$\nabla^{2}u = u_{xx}+u_{yy}=0$$
What name is given to solutions of Laplace's equation?
They are called harmonic functions.
When separating variables $u=X(x)Y(y)$ in $u_{xx}+u_{yy}=0$, what ODEs result?
$\frac{X''}{X}=-\frac{Y''}{Y}=\lambda$ (constant), giving $X''-\lambda X=0$ and $Y''+\lambda Y=0$.
Show the other 31 Partial Differential Equations flashcards
For a rectangular plate with $u=0$ on the sides $x=0,x=a$, what is the form of the $x$-eigenfunctions?
$X_n(x)=\sin\!\left(\frac{n\pi x}{a}\right)$ with eigenvalues $\lambda_n=\left(\frac{n\pi}{a}\right)^{2}$, $n=1,2,3,\dots$
State the mean value property of harmonic functions.
The value of a harmonic function at the centre of any circle (or sphere) equals the average of its values over that circle (sphere) boundary.
State the maximum principle for Laplace's equation.
A non-constant harmonic function attains its maximum and minimum values only on the boundary of the domain, not in the interior.
What is a Dirichlet boundary condition versus a Neumann boundary condition?
Dirichlet specifies the value of $u$ on the boundary; Neumann specifies the normal derivative $\frac{\partial u}{\partial n}$ on the boundary.
Write the Laplace equation in polar coordinates $(r,\theta)$.
$$\nabla^{2}u = u_{rr}+\frac{1}{r}u_r+\frac{1}{r^{2}}u_{\theta\theta}=0$$
Separating $u=R(r)\Theta(\theta)$ in polar Laplace equation, what ODEs arise?
$\Theta''+n^{2}\Theta=0$ and the Euler equation $r^{2}R''+rR'-n^{2}R=0$, with separation constant $n^{2}$.
What is the radial solution $R(r)$ of the polar Laplace equation for separation constant $n^{2}$ (with $n\geq1$)?
$R(r)=A r^{n}+B r^{-n}$; for $n=0$, $R(r)=A+B\ln r$.
Why is periodicity $\Theta(\theta)=\Theta(\theta+2\pi)$ imposed in polar problems, and what does it force?
Single-valuedness of $u$ requires period $2\pi$, forcing the separation constant to be $n^{2}$ with integer $n=0,1,2,\dots$ and $\Theta=a\cos n\theta+b\sin n\theta$.
Give the general (Fourier) solution of Laplace's equation inside a disk of radius $a$ (bounded at $r=0$).
$$u(r,\theta)=\frac{a_0}{2}+\sum_{n=1}^{\infty} r^{n}\left(a_n\cos n\theta + b_n\sin n\theta\right)$$ where negative powers and $\ln r$ are dropped for boundedness.
For the exterior problem ($r>a$, bounded at infinity), which radial terms are retained?
Only the decaying terms $r^{-n}$ (and the constant) are kept; the growing $r^{n}$ terms are discarded.
State Poisson's integral formula for the value of a harmonic function at the centre of a disk of radius $a$.
$u(0)=\frac{1}{2\pi}\int_0^{2\pi} u(a,\theta)\,d\theta$, the average of boundary values (a special case of the mean value property).
Write the one-dimensional heat (diffusion) equation.
$$u_t = k\,u_{xx}$$ where $k>0$ is the thermal diffusivity.
Separating $u=X(x)T(t)$ in the heat equation, what ODEs result?
$\frac{T'}{kT}=\frac{X''}{X}=-\lambda$, giving $T'+k\lambda T=0$ and $X''+\lambda X=0$.
Why must the separation constant be negative ($-\lambda$ with $\lambda>0$) for a heat problem with fixed-zero ends?
A positive or zero constant gives non-decaying or unbounded $T(t)$ and no nontrivial $X$ satisfying $X(0)=X(L)=0$; only $-\lambda<0$ yields decaying $T(t)=e^{-k\lambda t}$ and sine eigenfunctions.
Give the series solution of $u_t=k u_{xx}$ on $0<x<L$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=f(x)$.
$$u(x,t)=\sum_{n=1}^{\infty} b_n \sin\!\frac{n\pi x}{L}\, e^{-k\left(\frac{n\pi}{L}\right)^{2} t}, \quad b_n=\frac{2}{L}\int_0^{L} f(x)\sin\frac{n\pi x}{L}\,dx$$
What happens to the temperature distribution of the heat equation as $t\to\infty$ (insulated/zero-end case)?
Each mode decays like $e^{-k(n\pi/L)^{2}t}$, so $u\to$ the steady state (here $u\to0$ for zero-Dirichlet ends); higher modes decay fastest.
What is the fundamental solution (heat kernel) of the 1-D heat equation on the infinite line?
$$u(x,t)=\frac{1}{\sqrt{4\pi k t}}\,\exp\!\left(-\frac{x^{2}}{4kt}\right)$$ representing the response to an initial point source.
Does the heat equation describe a forward or reverse smoothing process, and is it time-reversible?
It smooths/diffuses initial data forward in time and is irreversible (ill-posed backward in time).
Write the one-dimensional wave equation.
$$u_{tt}=c^{2}u_{xx}$$ where $c$ is the wave speed.
State d'Alembert's solution of the wave equation on the infinite line with $u(x,0)=f(x)$, $u_t(x,0)=g(x)$.
$$u(x,t)=\frac{1}{2}\big[f(x-ct)+f(x+ct)\big]+\frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds$$
What are the characteristic lines of the 1-D wave equation?
The lines $x-ct=\text{const}$ and $x+ct=\text{const}$, along which left- and right-moving waves propagate.
What is the domain of dependence of the point $(x,t)$ for the wave equation?
The interval $[x-ct,\,x+ct]$ on the initial line; only initial data there affect $u(x,t)$.
Give the separated series solution of $u_{tt}=c^{2}u_{xx}$ on $0<x<L$ with fixed ends $u(0,t)=u(L,t)=0$.
$$u(x,t)=\sum_{n=1}^{\infty}\sin\frac{n\pi x}{L}\left(A_n\cos\frac{n\pi c t}{L}+B_n\sin\frac{n\pi c t}{L}\right)$$
For the vibrating string with fixed ends, what are the natural (angular) frequencies of vibration?
$\omega_n=\frac{n\pi c}{L}$, $n=1,2,3,\dots$, with $n=1$ the fundamental and higher $n$ the overtones.
How are the coefficients $A_n$ and $B_n$ in the wave series determined from $u(x,0)=f(x)$, $u_t(x,0)=g(x)$?
$A_n=\frac{2}{L}\int_0^{L} f(x)\sin\frac{n\pi x}{L}\,dx$ and $B_n=\frac{2}{n\pi c}\int_0^{L} g(x)\sin\frac{n\pi x}{L}\,dx$.
Compare the heat and wave equations in terms of order in time and qualitative behavior.
Heat: first order in $t$, parabolic, dissipative/smoothing, irreversible. Wave: second order in $t$, hyperbolic, energy-conserving, reversible, finite propagation speed.
What is the two-dimensional heat equation, and what steady state does it reduce to?
$u_t=k(u_{xx}+u_{yy})$. At steady state $u_t=0$, it reduces to Laplace's equation $u_{xx}+u_{yy}=0$.
Why does the wave equation preserve sharp features while the heat equation does not?
The wave equation transports data along characteristics without dissipation (energy conserved), whereas the heat equation's exponential damping of high modes smooths out discontinuities.
What general solution form does d'Alembert give for $u_{tt}=c^{2}u_{xx}$ in characteristic coordinates?
$u(x,t)=F(x-ct)+G(x+ct)$, a sum of a right-moving and a left-moving wave with arbitrary $F,G$.
In the canonical reduction, what coordinate change makes the wave equation $u_{\xi\eta}=0$?
Setting $\xi=x-ct$ and $\eta=x+ct$ transforms $u_{tt}=c^{2}u_{xx}$ into $u_{\xi\eta}=0$, integrated to $u=F(\xi)+G(\eta)$.
What boundary/initial data make the wave, heat, and Laplace problems well-posed, respectively?
Wave (hyperbolic): two Cauchy/initial conditions $u,u_t$ plus boundary data. Heat (parabolic): one initial condition $u(x,0)$ plus boundary data. Laplace (elliptic): boundary conditions only (Dirichlet/Neumann), no time data.
GATE Mathematics Partial Differential Equations FAQ
What is in the GATE Mathematics Partial Differential Equations syllabus?
GATE Mathematics Partial Differential Equations is split into 4 chapters: Method of Characteristics for First Order Linear and Quasilinear Partial Differential Equations, Second Order Partial Differential Equations in Two Independent Variables: Classification and Canonical Forms, Method of Separation of Variables for Laplace Equation in Cartesian and Polar Coordinates and Heat and Wave Equations in One Space Variable. Together they contain 6 topics.
How many chapters are in GATE Mathematics Partial Differential Equations?
4 chapters, containing 6 topics. The largest chapter is Method of Separation of Variables for Laplace Equation in Cartesian and Polar Coordinates, with 2 topics.
Are there flashcards for GATE Mathematics Partial Differential Equations?
Yes — a 51-card Partial Differential Equations deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.