# Partial Differential Equations

How can we prove (T (t)f)(s) := U(s, s − t)f(s − t) is a strongly continuous semigroup?
That is, how can it satisfy the conditions T(0) = I ,(T(t+s)= T(t)T(S) and continuity conditions?
Gul Hina · National University of Science and Technology
How do I find a Adomian polynomial of derivative terms?
NA=theta1*(del^2 theta/del x^2) NB= (del theta/ del x)^2
R. Mittal · Indian Institute of Technology Roorkee
For more detail you may see this link where ideas are correct but typographical mistakes http://www.veltechuniv.edu.in/IJMSC/current_issue/03_02-Paper06.pdf
What does "local supporting domain" mean ?
In avoiding the ill-conditioning problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within the local supporting domain.
How do I find stiffness matrix?
How do I find stiffness matrix when I know mass matrix, natural frequencies and mode shapes? Generally, If when know the stiffness and mass matrix, it is possible to find natural frequency and mode shape by doing eigenvalue analysis. So I think that is possible to find out the way that I said in the top line. I want to find a way by using MATLAB.
Huajiang Ouyang · University of Liverpool
Prof Link is right as a world authority on this topic. I feel there is one caution with the direct method ---- sometimes it does not give a physically realisable solution, that is,sometimes there are no mass/stiffness modifications to implement the solution of the direct method.
What is the best source for learning PDE (partial differential equations)?
Videos prefered.
Amro Elfeki · King Abdulaziz University
I think one of the effective way to learn is through applications you make yourself some simple codes once you know the theory. Below are some links (spreadsheet and ppt) among others available on the website that can help you do learn. Start with simple until you master it. https://www.researchgate.net/publication/256038858_One-Dimensional_Groundwater_Model_Details https://www.researchgate.net/publication/235955596_A_spreadsheet_model_to_solve_steady_state_groundwater_flow_equation_in_a_non-homogeneous_soil_column https://www.researchgate.net/publication/236949622_Simple_Groundwater_Model_on_Excel_Spreadsheet?ev=prf_pub https://www.researchgate.net/publication/236896993_Numerical_Solution_of_2D_Diffusion_Using_Explicit_Finite_Difference_Method?ev=prf_pub
Can someone share an hp fem matlab code for the singularly perturbed reaction diffusion equation in 2d with homogeneous boundary conditions?
Can anyone share the hp-fem matlab code for the 2d singularly perturbed reaction diffusion equation with homogeneous boundary conditions on (0,pi)^2.) −ε^2*Δu + a*u = f in Ω=(0,pi)^2, u = 0 on ∂Ω,
Adil AL-Rammahi · University Of Kufa
in MATLAB there is a solution for the finding the residues and poles and zeros. most frequency domain of (s,f(s)) is available.first you must read these information in matlab and second you can transform the problem in a form consist with matlab
How does an evolution semigroup (T(t)f) (s) = U(s, s-t)f(s-t) stasisfy the properties of semigroup?
The two conditions that are T(0) = I and T(s+t) = T(t)T(s)
Zeev Sobol · Swansea University
Well, first property is straightforward if you assume U to satisfy the property that U(s,s) = Id, and the second property requires U(t,τ)U(τ,s) = U(t,s) for t>τ>s. Indeed, T(t)T(τ) f (s) = T(t) [ U(s,s-τ) f(s-τ) ] = U(s,s-t) U(s-t, s - t - τ) f(s- t - τ) = U(s, s - t - τ) f(s- t - τ) = T(t+τ) f(s) However, this is just a beginning. The question on strong continuity is much more interesting, and a perturbation theory is a delight. See, e.g., a paper by Liskevich, Voigt and Vogt.
What is the most ill-conditioned kind of differential equation?
Based on this question: https://www.researchgate.net/post/What_are_the_more_robust_methods_for_solving_ill-conditioned_problems_linear_or_nonlinear
Adil AL-Rammahi · University Of Kufa
Bessel equation for its applications
What are the minimum requirements to perform GPU programming on a Laptop ?
Pankaj Mishra · IIT Kharagpur
Thank you Everyone :)
How to estimate the external force for the navier-stokes equation?
Firstly, we construct a map from [0,π] into [0,1], to introduce the function x(t), by immediate calculation we can get an estimate with constant C0 and C1. Then we un-determine the coefficient to search the value of C0,C1 so that we can estimate the operator function d(x), then to estimate the external force. In our study of the operator function d(x), we discover that under some assumption we can get an unique bound for the force f, which is formed by the constant c2. (page 10-16)
Aria Tsam · Aristotle University of Thessaloniki
Can we approximate to the singular point in analytic function of the klein-gordon equation?
if we can use the set to study the analytic function, can we construct an equivalent set,for example the set : which is equivalent to the set W, to achieve the topological equivalence which can omit the infinitely small and lead that we can get the the upper bound for the integral equation. to the cos case ,the upper bound is 1+i/2; to the sin case ,the upper bound is i+1/2i. if we can approximate to the singular point, what is this symmetry imply? how to apply this symmetry to the problems we meet in the klein-gordon equation? page13-17
Cheng Tianren · South China Agricultural University
Yes,but i don't use the traditional method.i approximate to the singular point by construct an equivalent set, and i found that the condition for approximation is related with a topological transformation. Page 17 figure 1
How can I get the criterion for the initial viscosity and the limit viscosity in the Navier-Stokes equation?
We used a well known integral inequality in the Navier-Stokes equation to get a criterion about the limit viscosity, the initial velocity, and the initial viscosity. Our procedure was: (1) Get an estimate for the velocity which is based on the parameter 2k0(1/4π), (2) relate the laplacian operator (for the velocity) and the M (the integral kernel of the viscosity), (3) estimate the external force in which we got a unique bound for the force f, and the bound was c2, which means we can use the bound to omit the surplus terms in the inequality which we mentioned at the beginning, and (4) use the parameter 2k0(1/4π)4 to study the relation among the limit viscosity and the initial velocity and initial viscosity.
F. Pla · University of Castilla-La Mancha
Depending of your application. If you consider geophysical models in which equations of motion are applied, you must looking for paper in which are taken viscosity and initial velocity values into account. Is very important consider real viscosity parameter values. For instance, in petrology models for lava dome volcans viscosity is in 10^6-10^15 Pa s range. I recommend you look for tables of values in those papers relationed with that research in which you are. I know that my answer is not fixed but the most important is my suggestion.
How to use identity representation and volume to study the relationship between viscosity and eigenvalue when using the Navier-Stokes equation?
We are studying the symmetric solution of the Navier-Stokes equation. First we get that under a condition, the viscosity converges to ∞ and the eigenvalue converges to ∞ . This condition is related to the function that u2+3/2u−(4γ−3)/4=0, in which we use the isentropic property of the Navier-Stokes equation and we also consider the condition of the delta Δ for the eigenvalue. Then we get the inequality about the identity representation and the volume by applying the property of the symmetric solution, such that: ∥1Q2Q∥≥1. Our problem is in how to use the inequality ∥1Q2Q∥≥1 to study the relationship between the viscosity and the eigenvalue.
Cheng Tianren · South China Agricultural University
Thank you very much! I will try to read these papers when i finish my current research!
• Cheng Tianren asked a question:
Can we use the symmetry in L(3,∞) space to study the velocity ratio of the navier-stokes equation?
In the navier-stokes equation, the Ls case such that s is included in (3,∞) is studied by ladyzhenskaya. In our research, we discover a new symmetry in the Ls case that when the parameter C is equal to [(1/2)3/2]/10,the L3 space holds; otherwise when the parameter C is equal to −[(1/2)3/2]/10, the L infinite space holds; our problem is how to apply this symmetry. A practicable way we propose is that we can use the velocity ratio to construct the Ls type space. And our method is based on an inequality that the integral of the velocity ∫v2 is less that v2, eventually we get this inequality by using the hermit polynomial.
• Cheng Tianren asked a question:
Any thoughts on the convergence of the series in the klein-gordon equation?
Here, I list a new method to study the convergence of the series in the klein-gordon equation. Firstly, transform the equation to the form Cj/(kj+ka)e−(kj+ka)x by analyze the matrix. Then, construct integral kernel to make it suit the form we get in the first step and also suit the standard form of the integral kernel equation. Finally, by comparing the value of C=Ym+1 we can determine the convergence of this type of equation in which we use the chebshev inequality to get the value C. Is this method feasible?
• Cheng Tianren asked a question:
Is this velocity estimate for weak solution in the Navier-Stokes equation useful?
Here, we raise a new method to estimate the velocity for weak solution.our procedure is: (1) Study the laplacian operator to get the condition that relate the time t and M (the integral kernel of the viscosity) and also prepare to transform the two-dimension coordinate to three dimension coordinate, (2) transform the equation to poisson form and also shift the coordinate to set the initial radius equal to zero and get the last estimate, in which we use a lemma in integral equation so that we can apply the spherical coordinate to estimate the value of cosa in the poisson solution. Our last estimate for the velocity is based on a parameter such that 2k0(1/4π)4, which is a valuable result that can be used in fluid mechanics (k0 is a max function about the initial velocity and the initial viscosity). Another technique problem we mentioned is that we can use the well known lemma in integral equations (which we used in step 2) to get a more practical condition of the velocity estimate need, which is a relation among the time t and the initial velocity and the initial viscosity.
Pseudo - differential operators. How do I calculate this?
What is the order the following pseudo differential operator R=-i \Delta +i \Delta J^{-1} where Jv=( I + i ( I-\Delta) )^{-1} v? How do I calculate this?
Morteza Koozehgar Kalleji · University of Mazandaran
the order of operator $\Delta$ is 2. Hence, one can conclude that the order of operator $J$ is also 2. Since operator $J$ is elliptic, so it has a parametrix operator ,$J^(-1)$, which is an order -2. Then composite of these two operator has order 0. If we consider R_{1} = \Delta and R_{2}= \Delta J^(-1), then according to asymptotic expansion theorem for symbols, we can imply R = R_{1} + R_{2} \in S^{max\{0,2\}) = S^{2}
What are the advantages of numerical method over analyatical method?
We use several numerical methods. Why do we use it and is it really applicable?
Petar Mali · University of Novi Sad
See, for example, this paper http://arxiv.org/pdf/cs/0701192v5.pdf
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.
Ekaterina Shevkoplyas · Saint Petersburg State University
Dear Albert, Thank you, I see the point and I'll think about that. However, the problem occurs not only in this case. I attached the description of the optimization problem in which we maximize the total payoff. It seems that nothing is missed, but there are still several solutions.
What is the most frequently used orthogonal polynomial over [-1,1]?
[-1,1] is the classical finite interval on which ops are defined.
Mahmood Dadkhah · Payame Noor University
1. 1Th chebychev Polynomial 2. Legendre Polynomial 3. B splines(but they should be orthogonalized by gram Schmidt procedure and fit to [-1,1])
What are the best perturbation methods for nonlinear PDEs ?
An example would be the nonlinear heat diffusion equation
Chinedu Nwaigwe · The University of Warwick
Try looking for the variational iteration method(VIM. It is interesting and fairly easy to understand. And could use help you understand other methods, such as the homotopy methods, with ease.
Does anyone have an example of a periodic function in the space $L^2(R^N)\cap L^\infty(R^N)$ ?
It will be very helpful in finding large energy solutions of some problems with lack of compactness.
Vladislav Babenko · Dnepropetrovsk National University
As far as I understand Francesco Di Plinio and A. Nandakumaran are right
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*).
Aria Tsam · Aristotle University of Thessaloniki
Good Evening. Sorry I dont know
Coupled PDE's with one is not homogeneous?
I used Galerkin's method to solve a coupled PDE'S, I assumed a solution of the form exp( λ), in order to convert the DE'S into polynomial of λ. one of the equations is not homogeneous and the term exp( λ) in the 2nd equation did not cancel out, can any body help me regarding this issue
Aria Tsam · Aristotle University of Thessaloniki
Good evening. try to read the sites: 1-http://www.me.metu.edu.tr/courses/me582/files/PDE_Introduction_by_Hoffman.pdf 2-http://www.ucl.ac.uk/~ucahhwi/LTCC/sectionA-firstorder.pdf