Questions without answers in Partial Differential Equation (PDE)

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:$(x+y)^{5}$$(x...

​​​​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 $T: T(x, y)$,...

The steady-state temperature distribution in a square plate $\mathrm{ABCD}$ is governed by the $2$-dimensional Laplace equation. The side $\mathrm{AB}$ is kept at a tempe...

A $5 \mathrm{~cm}$ long metal rod $\mathrm{AB}$ was initially at a uniform temperature of $T_0{ }^{\circ} \mathrm{C}$. Thereafter, temperature at both the ends are mainta...

The following function is defined over the interval $[-L, L]:$ $$ f(x)=p x^{4}+q x^{5} .$$If it is expressed as a Fourier series, $$f(x)=a_{0}+\sum_{n=1}^{\infty}\left\{a...

The function $f(x,y)$ satisfies the Laplace equation $$\bigtriangledown^{2}f(x,y) = 0$$on a circular domain of radius $r=1$ with its center at point $P$ with coordinates$...

The concentrations $s(x,t)$ of pollutants in a one-dimensional reservoir at position $x$ and time $t$ satisfies the diffusion equation$$\frac{\partial s (x,t)}{\partial t...

Consider the following expression: $$z = \sin \left ( y + it \right ) + \cos \left ( y - it \right )$$ where $\text{z, y}$ and $t$ are variables, and $i = \sqrt{-1}$ is a...

In the following partial differential equation, $\theta$ is a function of $t$ and $z$, and $D$ and $K$ are functions of $\theta$$$D(\theta)\frac{\partial^2\theta}{\partia...

The following partial differential equation is defined for $u:u (x,y)$$$\dfrac{\partial u}{\partial y}=\dfrac{\partial^2 u}{\partial x^2}; \space y\geq0; \space x_1\leq x...

The Fourier series to represent $x- x^2$ for $-\pi\leq x\leq \pi$ is given by$$ x-x^2 = \dfrac{a_0}{2} + \sum_{n=1}^{\infty} a_n\ \cos nx + \sum_{n=1}^{\infty} b_n\ \si...

Consider the following second-order differential equation: $y’’ – 4y’+3y =2t -3t^2$. The particular solution of the differential solution equation is$ – 2 -2t-t^2$$ – 2t-...

Consider a two-dimensional flow through isotropic soil along $x$ direction and $z$ direction. If $h$ is the hydraulic head, the Laplace’s equation of continuity is expre...

A one-dimensional domain is discretized into $N$ sub-domains of width $\Delta x$ with node numbers $i=0,1,2,3, \dots , N$. If the time scale is discretized in steps of $\...

The Fourier series of the function,$\begin{array}{rll} f(x) & =0, & -\pi < x \leq 0 \\ {} & =\pi – x, & 0 < x < \pi \end{array}$in the interval $[- \pi, \pi ]$ is$f(x) = ...

The type of partial differential equation $\dfrac{\partial ^2 P}{\partial x^2} + \dfrac{\partial ^2 P}{\partial y^2}+3 \dfrac{\partial ^2 P}{\partial x \partial y}+ 2 \df...

The solution of the partial differential equation $\dfrac{\partial u}{\partial t} = \alpha \dfrac{\partial ^2 u}{\partial x^2}$ is of the form$C \: \cos (kt) \lfloor C_1 ...

Consider the following partial differential equation:$$3 \frac{\partial ^2 \phi}{ \partial x^2} + B \frac{ \partial ^2 \phi}{\partial x \partial y} + 3 \frac{\partial ^2 ...

Laplace equation for water flow in soils is given below. $$\dfrac{\partial ^2H}{\partial x^2} + \dfrac{\partial ^2H}{\partial y^2} + \dfrac{\partial ^2H}{\partial z^2} = ...