Questions
1 question per paper
Difficulty
Medium
Importance
Medium yield for NTPC/ONGC/BHEL
Overview
Partial Differential Equations (PDEs) involve functions of multiple independent variables and their partial derivatives. For PSU exams, mastering the classification and standard solutions for physical phenomena like heat flow and wave propagation is essential for securing quick marks.
Classification of PDEs
Second-order linear PDEs are classified based on the discriminant b^2 - 4ac. This determines the nature of the solution and the physical behavior of the system described.
- Hyperbolic if b^2 - 4ac > 0
- Parabolic if b^2 - 4ac = 0
- Elliptic if b^2 - 4ac < 0
- Linearity depends on the dependent variable and its derivatives appearing to the first power
Laplace Equation
The Laplace equation describes potential fields or steady-state equilibrium. It is a fundamental elliptic PDE appearing frequently in electrical and thermal problems.
- 2D Laplace: del^2 u = 0
- u_xx + u_yy = 0
- Solutions are harmonic functions
- Governs steady-state temperature distribution
Heat and Wave Equations
These equations represent time-dependent physical processes. The Heat equation describes diffusion, while the Wave equation models mechanical oscillations or electromagnetic propagation.
- 1D Heat Equation: u_t = c^2 u_xx
- 1D Wave Equation: u_tt = c^2 u_xx
- c^2 is the thermal diffusivity in heat equation
- c^2 represents square of wave velocity in wave equation
Method of Separation of Variables
This is the standard analytical technique to solve PDEs by assuming the solution is a product of single-variable functions. It effectively converts the PDE into a set of Ordinary Differential Equations (ODEs).
- Assume u(x,t) = X(x)T(t)
- Equate to a separation constant (-k)
- Apply boundary conditions to determine the constants
- Use Fourier Series for the superposition of solutions
Formula Sheet
Hyperbolic: u_tt - c^2 u_xx = 0
Parabolic (Heat): u_t - c^2 u_xx = 0
Elliptic (Laplace): u_xx + u_yy = 0
Poisson Equation: u_xx + u_yy = f(x,y)
General 2nd Order: A u_xx + B u_xy + C u_yy + D u_x + E u_y + F u = G
Exam Tip
Focus on memorizing the standard form of the equations; most PSU questions ask you to identify the equation type rather than solving complex boundary value problems.
Common Mistakes
- Confusing the sign of the separation constant during the transformation into ODEs
- Misidentifying the type of PDE (Hyperbolic/Parabolic/Elliptic) due to algebra errors in the discriminant
- Failing to apply boundary conditions in the correct order
More Revision Notes
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