GATE Mathematics Partial Differential Equations Syllabus

Partial Differential Equations syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Partial Differential Equations in GATE Mathematics, not a summary of it.

1. 1

   Method of Characteristics for First Order Linear and Quasilinear Partial Differential Equations

   1 topic

   • Method of Characteristics
2. 2

   Second Order Partial Differential Equations in Two Independent Variables: Classification and Canonical Forms

   1 topic

   • Classification and Canonical Forms
3. 3

   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
4. 4

   Heat and Wave Equations in One Space Variable

   2 topics

   • Heat Equation
   • Wave Equation

Partial Differential Equations flashcards for GATE Mathematics

24 of 51 cards from the Partial Differential Equations deck — real questions with worked answers.

1. 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.

2. 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.

3. 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 $F$.

4. 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 $f$.

5. 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.

6. 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.

7. 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 $x,y$.

8. 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.

9. 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 $B^{2}-4AC<0$.

10. 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.

11. 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.

12. Classify the Laplace equation $u_{xx}+u_{yy}=0$.

    $A=1,B=0,C=1$, so $B^{2}-4AC=-4<0$: it is elliptic.

13. 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 $u_{\alpha\alpha}-u_{\beta\beta}=\dots$).

14. 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.

15. What is the canonical form of an elliptic equation?

    Using $\alpha,\beta$ coordinates it becomes $u_{\alpha\alpha}+u_{\beta\beta}=\Phi(\alpha,\beta,u,u_\alpha,u_\beta)$.

16. 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. $A\left(\frac{dy}{dx}\right)^{2}-B\frac{dy}{dx}+C=0$.

17. 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).

18. What is the Laplace equation in two-dimensional Cartesian coordinates?

    $$\nabla^{2}u = u_{xx}+u_{yy}=0$$

19. What name is given to solutions of Laplace's equation?

    They are called harmonic functions.

20. 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 $Y''+\lambda Y=0$.

21. 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 $\lambda_n=\left(\frac{n\pi}{a}\right)^{2}$, $n=1,2,3,\dots$

22. 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.

23. 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.

24. 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.

See more Partial Differential Equations flashcards →

Planning Partial Differential Equations for GATE Mathematics

Partial Differential Equations (GATE Mathematics) FAQ