Discussion
The partial differential equation \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^{2} u}{\partial x^{2}}\), where \(\alpha\) is a positive constant, is
Answer: D
The equation \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^{2} u}{\partial x^{2}}\) is parabolic.
A second-order equation \(A u_{xx} + B u_{xt} + C u_{tt} + \ldots = 0\) is classified by the discriminant \(B^{2} - 4AC\). Here \(A = \alpha\), \(B = 0\) and \(C = 0\), since there is no second derivative in \(t\), so
\[ B^{2} - 4AC = 0 - 4\alpha \cdot 0 = 0 \]
A zero discriminant means parabolic. This is the heat (diffusion) equation, the standard parabolic example. Elliptic needs a negative discriminant (Laplace's equation) and hyperbolic a positive one (the wave equation). 'Circular' is not a class of partial differential equation.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development