Partial Differential Equation (PDE) - Engineering Mathematics - Civil Engineering Previous Year Questions

The type of partial differential equation \(\frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2} + 3 \frac{\partial^2 p}{\partial x \partial y} + 2 \frac{\partial p}{\partial x} - \frac{\partial p}{\partial y} = 0\) is

The solution of the partial differential equation \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\) is of the form

The following partial differential equation is defined for \(u: u(x,y)\): \( \frac{\partial u}{\partial y} = \frac{\partial^2 u}{\partial x^2} ; \quad y \ge 0; \quad x_1 \le x \le x_2 \)
The set of auxiliary conditions necessary to solve the equation uniquely, is

The function \( f(x, y) \) satisfies the Laplace equation
\[ \nabla^2 f(x, y) = 0 \]
on a circular domain of radius \( r = 1 \) with its center at point P with coordinates \( x = 0, y = 0 \). The value of this function on the circular boundary of this domain is equal to 3.
The numerical value of \( f(0, 0) \) is:

The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of 100 °C and the other three sides are kept at a temperature of 0 °C. Ignoring the effect of discontinuities in the boundary conditions at the corners, the steady-state temperature at the center of the plate is obtained as T₀ °C. Due to symmetry, the steady-state temperature at the center will be same (T₀ °C), when any one side of the square is kept at a temperature of 100 °C and the remaining three sides are kept at a temperature of 0 °C. Using the principle of superposition, the value of T₀ is __________ (rounded off to two decimal places).

A 5 cm long metal rod AB was initially at a uniform temperature of \(T_0\,^{\circ}C\). Thereafter, temperature at both the ends are maintained at \(0\,^{\circ}C\). Neglecting the heat transfer from the lateral surface of the rod, the heat transfer in the rod is governed by the one-dimensional diffusion equation \(\frac{\partial T}{\partial t} = D \frac{\partial^2 T}{\partial x^2}\), where \(D\) is the thermal diffusivity of the metal, given as \(1.0\text{ cm}^2/\text{s}\). The temperature distribution in the rod is obtained as \(T(x,t) = \sum_{n=1,3,5,...}^{\infty} C_n \sin\frac{n\pi x}{5} e^{-\beta n^2 t}\), where \(x\) is in cm measured from A to B with \(x=0\) at A, \(t\) is in s, \(C_n\) are constants in \(^{\circ}C\), \(T\) is in \(^{\circ}C\), and \(\beta\) is in \(s^{-1}\). The value of \(\beta\) (in \(s^{-1}\), rounded off to three decimal places) is ______________.

The second-order differential equation in an unknown function \(u: u(x,y)\) is defined as \[ \frac{\partial^2 u}{\partial x^2} = 2 \] Assuming \(g:g(x), f:f(y)\), and \(h:h(y)\), the general solution of the above differential equation is

For the following partial differential equation, \( x \frac{\partial^2 f}{\partial x^2} + y \frac{\partial^2 f}{\partial y^2} = \frac{x^2 + y^2}{2} \) which of the following option(s) is/are CORRECT?

A partial differential equation \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \] is defined for the two-dimensional field \(T: T(x, y)\), inside a planar square domain of size \(2\text{ m} \times 2\text{ m}\). Three boundary edges of the square domain are maintained at value \(T = 50\), whereas the fourth boundary edge is maintained at \(T = 100\).

The value of \(T\) at the center of the domain is

The partial differential equation that can be formed from \( z = ax + by + ab \) has the form (with \( p = \frac{\partial z}{\partial x} \) and \( q = \frac{\partial z}{\partial y} \))

A partial differential equation is given below.
\[\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0\]
Possible solution(s) is/are: