Solving System of PDEs: Analytical Methods & Solutions

[satu]

TL;DR Summary

Methods to find solutions of system of PDEs

HI
HI! While trying to solve problem in Hydrodynamic stability I have got a system of Two Partial Diffential equations :

Can anyone help me to solve this analytically? Is there any general method to solve system of PDEs?

 

[renormalize]

Can you clarify your notation? E.g., does $\theta_{1x},\psi_{1x}$ mean $\partial\theta_{1}/\partial x,\partial\psi_{1}/\partial x$?

 

[wrobel]

nothing about boundary conditions
nothing about the domain
nothing about functional spaces
nothing to speak about

Is there any general method to solve system of PDEs?

no general methods to solve everything

 

[satu]

Can you clarify your notation? E.g., does $\theta_{1x},\psi_{1x}$ mean $\partial\theta_{1}/\partial x,\partial\psi_{1}/\partial x$?

Yup. You may consider $\psi$ inplace of $\psi_1$ and $\theta$ in place of $\theta_{1}$

 

[satu]

nothing about boundary conditions
nothing about the domain
nothing about functional spaces
nothing to speak aboutno general methods to solve everything

B.C's : $\psi_{1} = \theta_{1} = 0$ at both $y=0$ and $y=1$.
$\nabla^2 = \partial^2/\partial x^2 + \partial^2/\partial y^2$
Both $\psi_{1}$ and $\theta_{1}$ are functions of x and y.

 

[wrobel]

B.C's : $\psi_{1} = \theta_{1} = 0$ at both $y=0$ and $y=1$.

this does not make sense: your solutions are defined up to $+c_1 +c_2y$

 

[pasmith]

TL;DR Summary: Methods to find solutions of system of PDEs

HI
HI! While trying to solve problem in Hydrodynamic stability I have got a system of Two Partial Diffential equations :

View attachment 329594
Can anyone help me to solve this analytically? Is there any general method to solve system of PDEs?

This is a linearised stability problem, so you are looking for normal modes. Your assumption, in view of the boundary conditions, is that $$(\theta, \psi) = (\Theta e^{k_nx}\sin( n\pi y), \Psi e^{k_nx}\sin (n \pi y))$$ for constant $(\Theta, \Psi)$ and positive integer $n$. If $k_n$ has strictly positive real part, then the disturbance will grow as $x \to \infty$. If $k_n$ has strictly negative real part, the disturbance will instead grow as $x \to -\infty$.

Subsituting these into the PDEs will give you a linear system of the form $$M(k_n)\begin{pmatrix} \Theta \\ \Psi \end{pmatrix} = 0$$ for some matrix $M(k_n)$. For this to have non-zero solutions, $k$ must satisfy $$\det M(k_n) = 0.$$ (This is a generlized eigenvalue problem with $(\Theta, \Psi)$ being an eigenvector corresponding to the generalized eigenvalue $k_n$.)

The general solution, if desired, can then be obtained as a linear combination of normal modes, but for the linear stability analysis we only need the sign of the real part of $k_n$.

 

[wrobel]

anyway

linear stability analysis we only need the sign of the real part of $k_n$.

What definition of stability do you mean? As for me I heard about stability in evolution systems only. I do not see any evolution here and I do not see any uniqueness. I actually suspect that the boundary conditions still have not been formulated correctly but maybe x is such a new notation for time. Who knows.

 

[renormalize]

I actually suspect that the boundary conditions still have not been formulated correctly but maybe x is such a new notation for time. Who knows.

Indeed, it's unlikely that $x$ is supposed to be time since the OP states that $\nabla^2 = \partial^2/\partial x^2 + \partial^2/\partial y^2$, which is the 2D Laplacian, not the 2D wave-operator.

 

[pasmith]

anyway

What definition of stability do you mean? As for me I heard about stability in evolution systems only. I do not see any evolution here and I do not see any uniqueness. I actually suspect that the boundary conditions still have not been formulated correctly but maybe x is such a new notation for time. Who knows.

The OP did state "While trying to solve problem in Hydrodynamic stability ...". However, since the OP hasn't told us which problem they are trying to solve or how they derived this system, we can but speculate.

I did misinterpret the situation in my earlier post.

The system of PDEs looks like what one gets for marginal stability $(\partial_t = 0)$ of small disturbances to the static solution of the equations for incompressible Rayleigh-Benard convection in two dimensions, with $\psi$ being the streamfunction and $\theta$ the departure of the (scaled) temperature from the static solution; $R_0$ is then the Rayleigh number.

The correct approach is to first set $(\theta, \psi) = (\Theta(y), \Psi(y))e^{ikx}$ to obtain a system of ODEs for the dependence on $y$; we are looking for the wavenumber $k$ at which the static solution first becomes unstable - hence the assumption of marginal stability.

We are missing some boundary conditions - we need either $\psi_y = 0$ (no-slip) or $\psi_{yy} = 0$ (stress-free) at each boundary.