# Partial Differential Equations

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.

Ali Arshad · COMSATS Institute of Information Technology

Thanks@ Daniel for the concern shown, I will have to dig deep in the problem to get the solution. I have worked with almost all the variable and fixed step, stiff and non stiff ODE solvers but have not solved DAEs. My real task is to remodel a distributed system represented by a partial differential equation with DAEs, and then ultimately designing a sliding mode controller for the system

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

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.

Olena O. Vaneeva · National Academy of Sciences of Ukraine

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++/

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.

Chen Huyuan · NYUShanghai, Shanghai, China

Thanks, I got it.

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

Force can be substituted by curvature in general relativity, thus even if we do not use force, we use again a second order quantity.

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.

What refinement indicators currently exist for hyperbolic system of PDEs, in particular, the shallow water equations?

What refinement indicators currently exist for hyperbolic system of PDEs, in particular, the shallow water equations? which is your favourite and why? also provide references.

Chinedu Nwaigwe · The University of Warwick

Thanks Agah. I already have the paper and other related works from the first author.

I just want to know if there are other indicators in addition to weak local residual and numerical entropy production.

What's spectral stability?
What does it mean if a solution to a PDE is "spectrally stable"?
Vladimir Rasvan · University of Craiova

This problem has several issues. First, your solution should be considered in the setting containing both PDE and their boundary conditions. If all this stuff allows you to define a semi-group of operators along the solutions, the stability condition can be expressed in the language of the infinitesimal generator spectrum. This spectrum must lie in some half plane of C - the complex plane - defined by e.g. Re(z) less than some strictly negative real number. But the spectrum is not limited to eigenvalues but usually it contains also the continuum spectrum. Therefore - be careful!

Can anyone help with a heat equation, where boundary converges to the heat equation on the whole real line?

Dear All,

I have a, probably naive, question on the simple PDE:

For a one-dimensional heat equation with x in [-L, L] without external heat, boundary condition is u(t,-L)=0, u(t,L)=0 and initial condition is u(0,x) = 1 for x in [-0.5L, 0.5L], can we say that for any fixed T>0 the solution u(T, x)  of the above heat equation converges to u1(T,x) as the domain [-L, L] tends to [-infinity, infinity].

where u1(T,x) is the solution of a heat equation with the same initial value but on the whole real line.

If the answer is yes, could you please provide some references for it?

Many thanks

Yakov Krasnov · Bar Ilan University

How to solve non-linear differential equation using finite element method?

I want to use galerkin method to solve a nonlinear fourth order partial differential equation.The equation has 2 independent variables and its time dependent. I know how to come up with high order linear partial differential equation but have no idea of how to come up with non-linear ones.  I want to know how to form the element matrix for non-linear differential equation using galerkin method.Are there some specific books or references that talk about it?

Apostol Faliagas · Athens State University

The technique that is usually used to solve this kind of equations is linearization (so that the std finite element (FE) methods can be applied) in conjunction with a Newton-Raphson iteration. See for example how FE are used in freefem++ (www.freefem.org/ff++, manual, example 3.10 "Newton Method for the Steady Navier-Stokes equations") for the solution of the (non-linear) steady state Navier-Stokes equations. Another readily applicable source of example is http://fenicsproject.org/documentation/tutorial/nonlinear.html.

• Toka Diagana asked a question:
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)?

How to solve the set of differential equations if the stiffness matrix is singular?

My set of partial differential equations has a singular stiffness matrix. I use Matlab to solve this system. I found something used to find the inverse of stiffness matrix if it's singular which know as "pinv(A)". However, when I check the result, unfortunately it is not accurate. So, if anyone has a solution of my case I appreciate him with my thankful.

Thanks guys for interaction.

How can I implement numerical homogenization?
I am looking for articles or papers about implementation of numerical homogenization (two scale convergence.

Does anybody work with the Gerris Flow Solver?

To communicate about important difficult issue related to Gerris

Dan E Kelley · Dalhousie University

Yes, all sorts of people do.  If you state your actual problem you may get some help.

Which method (numerical or analytical) can I use to solve for eigenvectors and eigenvalues of a coupled partial differential equation?
a Uxx + b Uyy + c Uyx + d Ux + e Uy + f V = E1 U

g Vxx + h Vyy + k Vyx + m Vx + n Vy + q U^2 = E2 V.

All coeffients depensd on the variables x, y.

On the numerical side, I am exploring finite difference method and runge-kuta methods. But they seem not to give convincing results. Can not ensure othogonalisation of eigenvectors, etc..

I need both eigenvectors and eigenvalues to compute physical quantities, like conductivities, etc...

Can I also generalize the method for more than just two coupled systems?

Thank you.

The Bessel function in the Fourier-Bessel expansion is the Bessel function of the first kind. This has to do with the analytic property expected of the solution and the analytic property of the Bessel function of the first kind. Similarly as regards the Lagendre function that one encounters in the solution of the Schrödinger equation for Hydrogen; the choice of Pl(x) (the Legendre function of the first kind), to be contrasted with Ql(x) (the Legenrde function of the second kind), is related to the analytic property of Pl(x); the function Ql(x) is logarithmically singular at x=+/- 1. For details, consult any textbook on quantum mechanics. See also the book Handbook of Mathematical Functions, edited by Abramowitz and Stegun (Dover Publications).

• Sylvanus Kupongoh Samaila asked a question:
Can anyone show me how p-Laplacian and p(x)-Laplacian equations arise in electrorheological fluids, its origin and developments?

If the domain U is not smooth with wrought boundary, can I stil use Lebesgue-Sobolov space with a variable exponent?

How do you solve the two-dimensional eigenvalue problem in polar coordinates with homogeneous boundary conditions of the 3rd kind?
The boundary conditions at the outer radius of a disk are of the 3rd kind with spatially dependent coefficients
Victor F Petrenko · Dartmouth College
Use COMSOL 4.4 software. It's very easy to learn.
How can I answer this type of DE: u'(t)^2=4(u(t)-c)?
In the process of minimization a functional after putting Hamiltonian equation equal to a constant "c" I arrive at (x(t)^2)(1-x'(t))^2=c and I substitute u(t)=x(t)^2 and u'=2xx'
and obtain u'(t)^2=4(u(t)-c). I don't know the way of solving this?
Amaechi J. Anyaegbunam · University of Nigeria
Alireza Ahmadi wanted to solve the 1st order nonlinear
ODE x^2(1 – x')^2 = c --------------------- (1)
and proposed the transformation
u= x^2, u' = 2xx'. When this transformation is applied the
correct result is u '= 2[sqrt(u) +/- sqrt(c)] -------------- (2)
The transformed ODE (u')^2 = 4[u – c] ----------------- (3)
given by Ahmadi is wrong. This has been pointed out
in previous posts
Hence, it is incorrect to solve Eq. (3) as a prelude to
solving Eq. (1). As was also pointed out previously
no transformation is needed before solving Eq. (1)
How to solve a coupled 2nd order time-dependent PDE?
I want to solve analytically a coupled 2nd order space-time problem, originated from an optimal control problem. One of the problems is forward, another is backward in time. For example, (i) $y_t-y_{xx}=u, y(x,0)=0, y(0,t)=0, y(1,t)=g(t)$ (ii) $-p_t-p_{xx}=y, p(x,T)=0,p(0,t)=0, p(1,t)=h(t)$ with the coupling condition $p(x,t)+c*u(x,t)=0$ in $(0,1)\times (0,T)$. I have tried separation of variables, but it is getting complicated, any suggestions?
Bankim Chandra Mandal · University of Geneva
Actually not, I wanted to get analytical solution for this system. I checked that, people have worked in finding numerical solution, but I found very little about any kind of closed form analytical solution. Prof. Krzysztof Z. Sokalski has given a good technique to tackle these problems. Thank you for your consideration to this Q&A.