# Partial Differential Equations

Can anyone give a concrete physical interpretation of a complex conservation law except for conservation of momentum or energy?

I don't know the concrete physical interpretation of a complex conservation law which was constructed by some method, such as Noether's theorem.

Asher Klatchko · Reed College

What is a "complex conservation law"? How about the conservation of transport described by the "continuity equation"? \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0\, which when applied to energy would be interpreted as conservation of energy...

Does anyone know any physical processes governed by a PDE and nonlinear Robin boundary conditions?

The nonlinear Robin boundary condition is of the form :

$a \phi + b \frac{\partial \phi}{\partial n} + \phi \frac{\partial \phi}{\partial n} = c$,

where a, b, c may depend on time and space.

Qinghai Zhang · University of Utah

Antti, Thank you! This is a nice example.

Are matrices of Boundary Elements Method invertible?

I'm working on matrices arisen from collocation BEM, so I'm looking for a reference about them. Also I know matrices of Galerkin BEM is invertible but I don't know a way! Is there any draft about it?!

Vincenzo Mallardo · Universita degli studi di Ferrara

Actually to solve a system of equations means to invert it. Even LU factorization is a procedure that eventually makes the inversion of a matrix. I agree that if the problem is well-posed, the discretized problem has a solution

Does any have experience in modeling strategy of relaxation?

Given a PDE model with some constraints restricting the range of values of model quantities, how does one apply the idea of relaxation to such a model? Any idea or references on relaxation idea of modelling, will be appreciated.

Joao A. N. Filipe · University of Cambridge

Following on your reply re ODEs of 19 days ago. I focus on a very simple form of relaxation by assuming that the logistic equation, du/dt = a*u*(k-u), is related to your problem because its solutions do not exceed k if it is initially below k. Here, a, k > 0 are parameters, k is often known as ‘carrying capacity’. If we add an extra term with time-varying per capita rate v(t), we can be recast the equation as a logistic model with time-varying carrying and a thus new long-term bound on u. Specifically: du/dt = a*u*(k-u)+v(t)*u, with v(t)>0. This can be rewritten as: du/dt = a*u*((k+v(t)/a)-u) = a*u(q(t)-u), where q(t)=k+v(t)/a is a time varying carrying capacity. Unlike the basic logistic, this ODE is not likely to have an exact solution (unless v(t) is special, such as a constant), but numerical solution should be straightforward. There are many options for the function v(t), depending on the specific application. Here is an article exploring the case where q(t) also obeys a logistic equation; in this case, there is a form of relaxation if q increases with time.

P.S Meyer and JH Ausubel, Carrying Capacity: A Model with Logistically Varying Limits, Technological Forecasting and Social Change 61(3):209-214, 1999.

This specific choice of v(t) incorporates a time scale parameter characterising the pace of the relaxation, which may be relevant. This is just a concrete example to illustrate a way of thinking about modelling relaxation; many other basic models and modifications thereof to incorporate relaxation would be possible.

Sobolev extension

Let D be a bounded domain in RN. Is it possible to extend a function from H1(D) to H1(RN) without assuming the boundary of D to be lipschitz? I do not impose any requirements on the extension operator , e.g. to be continuous.

By H1(D) I mean the Sobolev space of square-integrable functions having their weak derivatives also square-integrable.

Maria Gokieli · University of Warsaw

Thanks!!! It is an interesting survey.

How to proceed with a non-unique solution of a HJB equation?
I get multiple solutions to the HJB equation which comes from discounted optimal control problem on infinite time horizon. I wonder if there are any ways to deal with such situation.
Any references or hints would be of great help.
Daniil Yurchenko · Heriot-Watt University

Thanks for sharing it.

Is there any Maple package for computing the approximate symmetries and conservation laws for a given system of PDE?

There is the GeM package for computing the symmetries and conservation laws for a given system of PDE.

Abdu Masanawa Sagir · University of Science Malaysia

Check for Maple 17 version

Can anyone help with a problem in solving a PDE using the finite difference explicit method?

I am solving the following energy equation:

1/alphamd* dT/dt =d2T/dz2 + f(z) where f(z) is some definite function, t is time, T is temperature, z is the unidimensional space variable and  alphamd is a parameter.

I used the finite difference explicit method to solve the eq. However for two different values of increments in time k and space h, I found different numerical solutions. I consider in both cases the same factor r=alphamd*k/h2 , both less than 0.5 (which is the limit for convergence of the method). Can anyone tell me why is this happening and what should I do in order to obtain a correct answer? Thank you very much

A. J. Roberts · University of Adelaide

I think you will have to give details of what BCs you want to apply, and how you have codes them into your Crank-Nicholson method.

• How to prove the x_N-odd solution of -\Delta u+|u|^pu=f in B_1, where f is x_N-odd?

We know that the solution of -\Delta u=f in B_1 is x_N-odd when f is x_N odd.

Бойков Владимир · National Research University of Information Technologies, Mechanics and Optics

All-purpose ofcgtrnbserver for any variety of models has been synthesized.

• What is the difference between the split-step finite difference method and just the finite difference method?
Could someone explain in brief and in a more understandable way the difference between the finite difference method and the split-step finite difference method?
Бойков Владимир · National Research University of Information Technologies, Mechanics and Optics

The method of integral measurement using for observer realization has been proposed. All-purpose ofcgtrnbserver for any variety of models has been synthesized.

• Do you think the Cooper pairs appearing in superconductivity could also be the magnetic monopoles had we used the symmetrized Maxwell's equation?

London had used Maxwell's equation to come to his known equations. What if we use instead the symmetric Maxwell equations encompassing magnetic monopoles? Can we obtain magnetic pairs (monopoles) as analogues for Cooper pairs?

Бойков Владимир · National Research University of Information Technologies, Mechanics and Optics

The task of state reconstruction of thermodynamic object based on its initial conditions reconstruction has been considered. The method of integral measurement using for observer realization has been proposed. All-purpose ofcgtrnbserver for any variety of models has been synthesized.

Can we ever be sure, that we have found all solutions of non-trivial Partial Differential Equation?

The PDE 2-nd degree (in general form) does not have the numerical method to solve it, does it?

J. W. Montgomery · University of North Texas

"Can we ever be sure, that we have found all solutions of non-trivial Partial Differential Equation?"

You have to define what you consider to be a solution before you can state whether or not you have found all solutions.

As R. C. Mittal said, the various types of PDEs are often paired with a certain type of conditions. The purpose of this is to characterize the set of solutions in terms of a seemingly simpler space. By this approach, one can say, "This is the set of all solutions which satisfy the following condition(s)."

"The PDE 2-nd degree (in general form) does not have the numerical method to solve it, does it?"

I do not know of a do-it-all method for 2nd-order PDEs. If limiting the class to linear PDEs then you can use a steepest descent method. The beauty of such methods is that you do not need to use conditions which yield existence and uniqueness and you can apply it to various types.

For example, if D is a given domain, then steepest descent in the Sobolev space H^2(D) can be used to solve
- Laplace's equation,
- the wave equation,
- the heat equation,
- Tricomi's equation,
- Keldysh's equation.
It is a type-independent method. All you really need is for the differential operator to be bounded in the space you descend in. Also, you can use a different inner product to generate the same Sobolev space and converge at a much faster rate. It is a versatile method, but other methods outperform it on specific problems, e.g.using a time-marching scheme for hyperbolic PDEs with initial-boundary value conditions.

Steepest descent can also be applied to nonlinear PDEs, but I cannot speak on whether or not it will work for an arbitrary PDE.

Why are most of the fundamental laws in Physics second order degree differential equations?
If we look at the laws of Newton, Schroedinger, Einstein and others we can observe that they are all second order degree differential equations, ordinary or partial. Why such a coincidence? Is this an indicator that our projection of reality is just a linear projection or is it something deeper behind this universality of the 2nd degree?
Demetris Christopoulos · National and Kapodistrian University of Athens

Dmitri, the concept of 'linear' here mainly means the superposition property: The differential operator in general is a linear one. Thus the so many applications of Fourier Analysis. Not the way that we can solve them.

How can we simulate Differential Algebraic Equations (DAEs) in MATLAB

One of the methods for developing control system for distributed parameter systems is to convert the system of PDEs in to a DAE; consisting of independent ODEs related to one another by algebraic equations or constraints.

Saman Rahbar · SSP Co.

Dear Ali,

ode15s solver can be a nice shot!!! but Matlab ODE helper can help you much more ,

Regards,

Can you suggest how to (analytically or numerically) solve this set of coupled PDE's?

Hello all,

I have a set of coupled PDE with well defined domain / boundary conditions, but it's beyond my knowledge to solve it. Could anyone suggest a good method to analytically or numerically solve it? Thanks in advance!

\left[ x \partial_x + y \partial_y + H(x)+ H(y) \right] T_0(x,y) ={}& 1+H(x) T_y(x,y) +H(y) T_x(x,y), \\
\left( 1- \partial_x \right )T_x ={}& T_0,\\
\left( 1- \partial_y \right )T_y ={}& T_0.\\

\\
\\
\text{Domain } \Omega = \left\{ (x,y) \vert x>x_0, y<y_0, x<y \right\}
\\
\\
\text{Boundary Conditions: }\\
T_0(x_0,y) = 0, \forall y \in (x_0,y_0)\\
T_x(x,x) = 0, \forall x \in (x_0,y_0)\\
T_y(x,y_0)=0, \forall x \in (x_0,y_0).

PS0. Try http://www.codecogs.com/latex/eqneditor.php to visualize the LaTex.
PS1. H(x) is the nonlinear repressing Hill function. Even if one can solve the limiting case $n\rightarrow \infty$ it will be interesting to me.

PS2. I know that this is equivalent to a 3rd order PDE... but it seems to be very complicated so I just presented the original 3 coupled first-order PDEs.

PS3. I've tried the finite difference method to integrate the PDEs, but the results do not seem to be satisfactory near $x=y$. It seems to me that some numerical instability is involved.

Zarghaam Rizvi · Christian-Albrechts-Universität zu Kiel

you can also try COMSOL...It is very good in solving algebraic and Partial differential eqns

Evolution family associated with the algebraic sum A(t) + B(t) - any thoughts?

Let A(t) and B(t) be unbounded linear operators on a Banach space X. Suppose the algebraic sum, A(t) + B(t), makes sense (nontrivial) and that A(t) and B(t) have evolution families associated with them, which we denote by U(t,s) and V(t,s). Under what conditions does A(t) + B(t) have an evolution family W(t,s)? In that event, what are the connections between the evolution families U(t,s), V(t,s), and W(t,s)?

Toka Diagana · Howard University

Thank you for your interest in this question. The case you are referring to is already known (see for instance the work done by P. R. Chernoff, M. Fuhrman, etc in this direction). You’ve probably noticed my question concerns unbounded linear operators A(t), B(t) which depend on a parameter t. (For a given family of linear operators A(t): D(A(t)) X → X, the existence of an evolution family ---- which in general is not the same as a semigroup --- associated with it is a very tricky question. Obviously, there are some cases where one has the existence of an evolution family U(t,s) associated with the family of operators A(t). That is for instance the case if the linear operators A(t) satisfy the so-called Acquistapace-Terreni conditions.)

Do Maxwell's equations alone lead to constant velocity of light in all inertial frames?

We normally think that the velocity of light is constant in all inertial frames is a postulate or the experimental result of Michelson-Morley experiment. But is it true that the Maxwell's equations lead to this and we do not need any extra postulate or any experimental results? Can anyone clarify this to me?

Can you suggest good references for learning Chaos in Partial Differential Equations?

While there are many papers and good books about chaos in ordinary differential equations, I like to know if there are some good books and survey papers about chaos in partial differential equations. Your suggestions are highly appreciated.

Best wishes

Toufic El Arwadi · Beirut Arab University

http://www.ams.org/journals/proc/2012-140-06/S0002-9939-2011-11069-4/

This is an example about Chaos for PDE

Can anyone help me find a FEM code to solve nonlinear partial differential equations for fluid flow?

Finite element method code in matlab or in mathematica to solve navierstokes equations for fluid flow

Mourad Ismail · University of Grenoble

http://www.feelpp.org/

http://www.freefem.org/ff++/

Do you have solved examples of system of nonlinear pde using finite element method?

See above

Is it possible to derive boundary integral equations for inertial flows?

Boundary integral formulations are considered as robust and efficient methods used to solve the linearized Navier-Stokes equation. The partial differential equations are transformed into integral equations by Green’s identities, where the velocity field is represented as a combination of hydrodynamic potentials of single and double-layer. BIM formulation could also be extended in order to solve for non-Newtonian fluids. I would like to know whether similar methods can be derived as well for the full NS equation where the nonlinear inertial forces cannot be neglected compared next to the linear viscous forces? Thank you.

Bernardo Figueroa · Universidad Nacional Autónoma de México

I am not sure, however it is possible to solve problems strongly dominated by inertia using the Boundary Element Method, here is a beautiful example:

Ha-Ngoc, H. & Fabre, J. 2004 Test-case No. 29B: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN) Part II: In a flowing liquid.
Multiphase Sci.Technol.16, 191–206

Solving coupled pdes for a viscoelastic cantilever?
I don't know how to deal with the complex damping E*. I am trying to study the dynamics of a cantilever subjected to bending and torsion, the beam is made of a viscoelastic material. After setting up the equations of motion and using Galerkin's method to convert the equations to odes in matrix form, I have a difficulty in the implementation of the viscoelastic property ( what to do with E* & G*).
Giovanni Bratti · Federal University of Santa Catarina

There is no problem in use E* instead of E. So you will be using a Structural Damping model, and you can solve your problem using "Direct Analysis".

You can see more details about it on the book:

Introduction to finite element vibrations analysis, Maurice Pety, ISBN 052126607-6, chapter 9.

What is the best way of numerically solving a system of nonlinear PDEs?

I have a system of 6 nonlinear PDEs. These equations involve a time derivative and one spatial derivative. What would be the simplest way to get a time dependent solution of these equations?

Klaus Schittkowski · University of Bayreuth

How do you numerically integrate time dependent exponentials?
In one of my problems I tried to numerically integrate the following function, F(t) = Exp(-0.5 * t).

Can we use Simpon's rule to integrate it? Or are any other methods used to numerically integrate F(t)?
Saeed Kazem · Amirkabir University of Technology

If you want to use the Simpon's rule to integrate, it's better to divide the domain of integration to m sub-domain and then apply the Simpson's rules for each sub-domain. Therefore the order of convergence for this method is O(h^3/m^2).

Why is a system with PDEs infinitely dimensional?

Partial Differential Equations (PDEs) contain at least two independent variables. Generally the system of PDEs is called infinite dimensional, what is the reason behind this argument?

Daniel Guan · University of California, Riverside

Let us say that you have two free variables x and t. Consider the initial value condition

at t=0 and x could be arbitrary at least locally. That is, you already know u(x, 0). Then you have an equation of t for each (fixed) x, if this is true just for an example. You could get a solution for each x.

Therefore, generically speaking, the solution is depended on the initial condition u(x, 0),

which is a function depended on x.

The solution space usually is only a kind of infinite dimensional variety (or manifold if it is kind of smooth). It is only a kind of linear space if the PDE is linear (this answer your second question).

Best,

DG

How is a weak solution of a partial differential equation usefull in Physics and Engineering?

In the last few years I always thought as an engineer that the solution a physical system 'produces' is always smooth (differentiable to a certain degree). These solutions are so called classical solutions.

But now I have learned of weak solutions that can be found for partial differential equations. Those solutions don't have to be smooth at all, i.e. they have to be square integrable or their first derivative must be square integrable ...

So, if the weak solution is not differentiable it will not satisfy the original differential equation. Now, what is the use of the weak solutions that can be found? What is their physical meaning and how are they useful to find classical solutions?

Peyman Hessari · Ulsan National Institute of Science and Technology

Weak solutions are easy to understand and implement, however they are not always the solution of the original PDEs.