Solutions of one dimensional wave equation.

Q. Solution of Laplace's equaiton having continuous second-order partial derivatives are called

1. biharmonic functions
2. harmonic functions
3. conjugate harmonic funcitons
4. error functions

Q. The number of boundary conditions required to solve the differential equation $\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}$ is

1. $2$
2. $0$
3. $4$
4. $1$

Q.

A one-dimensional domain is discretized into $N$ subdomains of width $\Delta x$ with node numbers $i=0,1,2,3,...,N$. If the time scale is discretized in steps of $\Delta t,$ the forward-time and centered-space finite difference approximation at $i^{th}$ node and $n^{th}$ time step, for the partial differential equation $\frac{\partial v}{\partial t}=\beta \frac{{\partial }^{2}v}{\partial x^2}$ is

1. $\frac{v_i^{\left(n\right)}-v_i^{(n-1)}}{2\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{2\Delta x}\right]$
2. $\frac{v_i^{\left(n\right)}-v_i^{(n-1)}}{\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{(\Delta x{)}^2}\right]$
3. $\frac{v_{i+1}^{(n+1)}-v_i^{\left(n\right)}}{\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{2\Delta x}\right]$
4. $\frac{v_i^{(n+1)}-v_i^{\left(n\right)}}{\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{(\Delta x{)}^2}\right]$