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# Partial Differential Equations - Science topic

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Questions related to Partial Differential Equations
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usually Floquet is the periodic boundary in solid mechanics module, but I need to use my equation in PDE module to construct the material, however, in pde, there is only periodic boundry but no Floquet, or, is it possible to set the material maually in local variables and put it in equations in the solid mechanics?
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Reduction of one PDE equation by Lie symmetry method and other method is always give the same result ?
In other methods, like the Kudryashov method, Riccati equation method,
trial function method and (G'/G)-expansion method, we reduce the PDE into ODE by travelling wave transformation. As a result, they provide only travelling wave solutions. On the other hand, Lie symmetry analysis gives us certain transformations, under which our PDE remains invariant.
Hence, the key advantage of Lie symmetry analysis over other analytic
methods is that it provides travelling solutions and in some cases, it
also provides non-travelling solutions.
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I first study finite element method and dealing with matlab program, I can solving PDE by linear trianglution mesh but I can't by quadratic or higher degree for triangulation. My question is: How I can make a matlab code for solving a PDE by Quadratic finite element method ?
Sorry!, I can’t give you detail information on this.
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Dear colleagues, we know that getting a new research paper published can be a challenge for a new researcher. It is even more challenging when considering the risk of refusal that comes from submitting a new paper to a journal that is not the right fit. we can also mention that some journals require an article processing charge (APC) but also have a policy allowing them to waive fees on request at the discretion of the editor, howover we underline that we want to publish a new research paper without APC!
So, what do you suggest?
We are certainly grateful for your recommendations. Kind regards! ------------------------------------------------------------------------------
Abdelaziz Hellal Mohamed Boudiaf M'sila, University, Algeria.
Cubo, a mathematical journal
Is very good for this fields of mathematics and more than it.
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Hello everyone,
Some discretizations of pressure-displacement formulation in Biot poroelasticity produce spurious pressure oscillations in some cases, for example, for small time step sizes. If I understood correctly, FVM for flow and P1-FEM for mechanics fall in this category. How does one rigorously prove existence of such instability? Is it possible to obtain any estimates on time step size for instability to occur? I guess this analysis can be carried out in FEM framework in some way.
Dear collegue,
I faced this issue some times ago. Same issue for transient thermal problems (since the equations are the same). Actually I didn't find rapidely what this came from, but fixed it by using a rule-of-thumb concerning a lower bound for the time step. You can find this (with references) in open document at https://hal.archives-ouvertes.fr/hal-00321789 (sentence just before section 3.2).
Note that this is not a problem of instability of the time integration scheme (that lead to an upper bound of the time step).
By coincidence, I found the basic theory underlying this phenomena called Principle of Discrete Maximum (by Ciarlet). This is a property that satisfies the continuum problem (without discretisation). The issue is that the discrete version may fail to satisfy it. Actually, it seems that another way to overcome this is to use lumped capacity matrices.
Hope this helps.
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i have attached some problem i m trying to do with Mathematica HAM for this kind equations
why you are not applying the newly developed Coiflet wavelet Homotopy analysis method to solve these PDEs. for details please read the work of Professor Liao and Yu Qiang
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Hello everyone,
I want to solve a elastic wave problem in Comsol using PDE module since my model have a different consititutive relation which is not included in the present modules.
I have difficulty when using the " Coefficient type partial differential equations" because I can't find "Frequency Domain" in research module. "Transient" "Eigenvalue" and "Steady" are available.
How can I calculate a frequancy domain problem with PDE in Comsol?
By the way, if "PML" can be realized in PDE?
I would be really appreciated it if you can help me find out.
Best Regards,
Hao Qiu.
You can see my video ant my channel
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A trace of Ciprofloxacin was measured in the product. Need to set the PDE to justify the measured amount is low in the product.
NOEL Value 1.2 mg/Kg
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Is it mandatory to calculate permitted daily exposure from human data as well as animal data and whichever PDE is low should be considered ?
It is necessary to calculate permitted daily exposureof human data as well as animal data because of the data can differ
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Hello,
Would you have an idea on how to model stable regular networks in a phase field model ?
For instance a square network like in the joined picture (the picture was made artificially) ? I know that the Swift-Hohenberg equation allows to create some regularity but I would like to control more the size of the network and the distance between the branches, and maybe the type of network.
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I want to model a photovoltaic panel that contains sublayers to obtain layers temperature changes over time. I have a bunch of formulas that discretize in terms of time and space. In other words, I want to solve a system of PDEs( each PDE is a discrete formula for each panel layer).
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I would like to start a discussion of this specific topic.
Here I would like to discuss the list of possible techniques helpful for performing the simulation of oscillating bodies in quiescent fluid.
This discussion is open to all the students, teachers, and researchers.
I request you to reply here if you are familiar with code development in OpenFoam, IBPM, NEEK1000, lilypad and CFX
Pradyumn Chiwhane I am posting here a previous answer I provided on submerged oscillating bodies in quiescent fluid. The total force exerted by the fluid on the cylinder, you should consider besides the drag the added mass force. Academic references are provided within the following research projects:
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1. Bose-Einstein condensation: How do we rigorously prove the existence of Bose-Einstein condensates for general interacting systems? (Schlein, Benjamin. "Graduate Seminar on Partial Differential Equations in the Sciences – Energy, and Dynamics of Boson Systems". Hausdorff Center for Mathematics. Retrieved 23 April 2012.)
2. Scharnhorst effect: Can light signals travel slightly faster than c between two closely spaced conducting plates, exploiting the Casimir effect?(Barton, G.; Scharnhorst, K. (1993). "QED between parallel mirrors: light signals faster than c, or amplified by the vacuum". Journal of Physics A. 26 (8): 2037.)
Regarding the first problem, there are many examples. For instance, the paper by O. Penrose, Bose-Einstein condensation in an exactly soluble system of interacting particles'', esearchportal.hw.ac.uk/en/publications/bose-einstein-condensation-in-an-exactly-soluble-system-of-intera
Cf. also, the paper by E. Lieb and R. Seiringer, Proof of Bose-Einstein Condensation for Dilute Trapped Gases'',
Regarding the second problem, the boundary conditions break Lorentz invariance. That's why the question isn't well-posed, whether in the classical limit or when quantum effects must be taken into account. In a finite volume it requires care to define the propagation velocity properly, since the equilibrium field configurations describe standing waves.
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I am trying to solve a time-dependent diffusion equation with finite-difference discretizations using the Newton–Raphson method. However, I encountered some problems in convergence--the solution result changes when changing time step size. Here I want to find some coding examples to refer to. Thank you.
In general, the name diffusion equation (time dependent or not) identifies a linear PDE, whose finite difference discretization leads to a linear algebraic problem that does not require a nonlinear solver like Newton Raphson. This means that your problem is a nonlinear one, for example with nonlinear viscosity (porous media equation and such), and that an implicit method is being used for the time discretization. The correct way to approach these problems is to write the space and time discretization (there is no mention of what time discretization method is used) which leads to a (potentially large) nonlinear system to be solved
at each time step. Then at each time step a nonlinear solver is used to compute the solution of thie algebraic problem. Any of these steps might have been coded incorrectly leading to the problem you mention. One simple way to check what is going on is to compare with the results of a simple explicit method (say forward Euler) employed with a very small time step, such a method would not require NR iterations and therefore any incorrectness in the NR solver would not affect it.
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I am trying to solve the differential equation. I was able to solve it when the function P is constant and independent of r and z. But I am not able to solve it further when P is a function of r and z or function of r only (IMAGE 1).
Any general solution for IMAGE 2?
Kindly help me with this. Thanks
check out this paper using a Laplace transformation for solving nonlinear nonhomogenous partial equations
the solutions are not trivial which is different if your coefficients and the pressure P are constant
Hopefully it helps
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Consider the powerful central role of differential equations in physics and applied mathematics.
In the theory of ordinary differential equations and in dynamical systems we generally consider smooth or C^k class solutions. In partial differential equations we consider far more general solutions, involving distributions and Sobolev spaces.
I was wondering, what are the best examples or arguments that show that restriction to the analytic case is insufficient ?
What if we only consider ODEs with analytic coeficients and only consider analytic solutions. And likewise for PDEs. Here by "analytic" I mean real maps which can be extended to holomorphic ones. How would this affect the practical use of differential equations in physics and science in general ? Is there an example of a differential equation arising in physics (excluding quantum theory !) which only has C^k or smooth solutions and no analytic ones ?
It seems we could not even have an analytic version of the theory of distributions as there could be no test functions .There are no non-zero analytic functions with compact support.
Is Newtonian physics analytic ? Is Newton's law of gravitation only the first term in a Laurent expansion ? Can we add terms to obtain a better fit to experimental data without going relativistic ?
Maybe we can consider that the smooth category is used as a convenient approximation to the analytic category. The smooth category allows perfect locality. For instance, we can consider that a gravitational field dies off outside a finite radius.
Cosmologists usually consider space-time to be a manifold (although with possible "singularities"). Why a manifold rather than the adequate smooth analogue of an analytic space ?
Space = regular points, Matter and Energy = singular points ?
For a function describing some physical property, when complex arguments and complex results are physically meaningful, then often the physics requires the function to be analytic. But if the only physically valid arguments and results are real values, then the physics only requires (infinitely) smooth functions.
For example, exp(-1/z^2) is not analytic at z=0, but exp(-1/x^2) is infinitely smooth everywhere on the real line (and so may be valid physically).
One place where this happens is in using centre manifolds to rigorously construct low-D model of high-D dynamical systems. One may start with an analytic high-D system (e.g., dx/dt=-xy, dy/dt=-y+x^2) and find that the (slow) centre manifold typically is only locally infinitely smooth described by the divergent series (e.g., y=x^2+2x^4+12x^4+112x^6+1360x^8+... from section 4.5.2 in http://bookstore.siam.org/mm20/). Other examples show a low-D centre manifold model is often only finitely smooth in some finite domain, again despite the analyticity of the original system.
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Suppose one has the following PDE in (x,y):
a(x,y)*(fxx + fyy) + b(x,y)*fx = e(f), (1)
where fx= partial f/partial x, etc, and
where a(x=1,y) = 0, so that along the line x=1, the PDE becomes
b(1,y)*fx = e(f). (2)
Does condition (2) effectively serve as a boundary condition on the original PDE (1) to solve in the region x <= 1?
I think there is something is missing.
This is not the right approach to create boundary conditions for the second-order PDE. For example:
If (x2 - 1)(fxx + fyy) + (x2 - y)fx = f2 (1)
According to your claim, If x=1, then
(1- y)fx = f2 is a boundary condition over the region x<= 1.
Which is not correct for many reasons.
One of them is that you substitute x =1 only for the coefficients a(x,y) and b(x,y). What about the solution f(x,y) itself? The study should include the effect of the solution and not only the coefficients involved!
Best regards
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We use transformations to convert the PDE into ODE to find the exact solutions of the PDE. If we use the different definitions of derivative in the transformations, then is it possible to obtain a different ODE of the given same PDE?
I agree with Dr Muhammad Ali
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I have the following Laplace's equation on rectangle with length a and width b (picture is attached):
ΔU(x,y)=0
Ux(0,y)=0    : Isolated boundary
Ux(a,y)=f(y) : Current source
U(x,b)=0      : Zero potential
The fourth boundary is quite complex :  Mixed boundary condition (isolated except in two points)
if x=a/3 or 2a/3
U(x,0)=0
else
Uy(x,0)=0
Is there an analytical way to solve this kind of mixed boundary problem? can someone point me to the right direction? I'm a bit lost
Finite difference methods using a discretization on Cartesian Grid (you can choose which type A, B or C Arakawa Grid) -- I think there are few tutorials on youtube
Finite Element methods (weak form) -- https://www.youtube.com/watch?v=U65GK1vVw4o&list=PLJhG_d-Sp_JHKVRhfTgDqbic_4MHpltXZ (really good tutorial with few examples and coding tutorials)
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Which numerical method is more effective for solving parabolic diffusion PDE in 2D [ with Dirichlet boundary conditions, initial condition at time t =0 is given]? The PDE may represent time dependent heat equation in two space variables, an IBVP.
Thanks!
Dear Tekle Gemechu,
Regarding this question, I am sharing my experience here in this area which may be useful.
I would suggest Crank-Nicolson method for solving your Diffusion PDE, due to the enhanced convergence properties through this method. Unfortunately, there is a small problem. As far as the heat transfer equation is concerned, by taking into account the dependency of thermal diffusivity (thermal properties) to the temperature, numerical modeling of the implicit part of the discretized Crank-Nicolson equation will not be so easy. Therefore, as my second suggestion, It is better to use an explicit time discretization and a simple spatial discretization (second-order central finite difference, as it is a diffusion problem).
Regards.
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I want to solve partial differential equations (PDEs), which contains both space (x) and time (t). What is the best software for this purpose?
I also want to know the most appropriate numerical algorithm for this so that I can write a program to solve PDEs.
All types of suggestions are highly appreciable.
Thank You.
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Hello!
Could anyone provide and advice regarding if its better to use in your PDE calculation the doses expressed in mg/m2 or if its better to convert them on a mg/kg basis. I saw that is possible to convert the dose expressed in mg/m2 to mg/kg by dividing the Animal Dose by Km value (according to FDA guidance for industry) , is this a correct approach?
I was wondering this because firstly i took in consider to multiply in the PDE formula the animal dose (mg/m2) from preclinical studies with average body surface for humans (1.62m2) before applying the adjustments factor, and i dont know if this or the previous one is the correct approach.
Hi
Generally, the BSA dosing paradigm has more than 50 years, and it's based on observations explained in FDA Guidance based on two papers:
"antineoplastic drugs, doses lethal to 10 percent of rodents (LD10s) and MTDs in nonrodents both correlated with the human MTD when the doses were normalized to the same administration schedule and expressed as mg/m2".
Allometric scaling (conversion) approaches suggested by different Authorities cover rather BW not BSA in relation to PDE. ICH, FDA, EPA, ECHA suggest BW0.75 or BW0.67 depending on species and approach. It is suggested that BW0.67 correlates more closely with surface volumes and BSA. See the Sussman paper below:
Best regards
Tomasz
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In the Reproductive toxicity test，Maternal general toxicity NOAEL:0.3 mg/m2.Embryofetal developmental toxicity NOAEL:0.45 mg/m2.Can I only choose embryo fetal toxicity NOAEL to calculate PDE？as follows。
Hello!
Could anyone provide and advice regarding if its better to use in your PDE calculation the doses expressed in mg/m2 or if its better to convert the doses on a mg/kg basis. I saw that is possible to convert the dose expressed in mg/m2 to mg/kg by dividing the Animal Dose by Km value (according to FDA guidance for industry) , is this a correct approach?
I was wondering this because firstly i took in consider to multiply in the PDE formula the animal dose (mg/m2) from preclinical studies with average body surface for humans (1.62m2) before applying the adjustments factor, and i dont know if this or the previous one is the correct approach.
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Many Monte Carlo methods to solve a given Partial Differential Equation (PDE) are built by sampling the PDE's Green's function. E.g., for heat diffusion, diffusion-convection-reaction type of equations, and so on, have algorithms that can be derived directly from the PDE (i.e., through Ito calculus or stochastic integral). On the other hand, for the Radiative Transfer Equation (RTE), there is an Integral representation. However, the argument for explaining Monte Carlo Radiative Transfer (MCRT) ALWAYS revolves around the physical interpretation.
I even found a review article [1] that states on page 16: "Unlike traditional approaches to RT problems, MCRT calculations do not attempt to solve the RTE directly."
Is there really NO relation (discovered yet) between MCRT and the RTE? or is it just that no one has ever proven this?. I understand the physical interpretation; it is just that having mathematical foundations would also help teach it in class. Can anyone help me by directing me to a reference that derives this?.
[1] Noebauer, U. M., & Sim, S. A. (2019). Monte Carlo radiative transfer. Living Reviews in Computational Astrophysics, 5(1), 1-103.
Stam Nicolis I checked these references thoroughly. Actually, the document written by Duncan Forgan was the one I used to implement (probably) my first Radiative Monte Carlo code about 10 years ago. Still, I couldn't find what I was looking for. Maybe I am not making myself clear, but what I would like to know is if there is a document that clearly and explicitly explains the relation between the MCRT algorithm (i.e. steps involved and the iterative structure) and the Radiative Transfer Equation (e.g. the algorithm's relation to each of the operators present within the equation). I.e. a relation between MCRT and the RTE without resorting to the "imagine what happens to a photon in the physical scenario".
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This paper is a project to build a new function. I will propose a form of this function and I let people help me to develop the idea of this project, and in the same time we will try to applied this function in other sciences as quantum mechanics, probability, electronics …
Are you sure you have defined your function correctly?
1. Usually z=x+iy. But in your function z is in the limit, thus being in both the arguments and what the integral is computed against. If z is not x+iy, the function is not a function of (x,y).
2. What do you mean by limit? Do you want to compute the case when z->0?
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Hi
I wanna solve partial differential equation in terms of x and t (spatial and time), As I know one of the most useful way for solving pde is variable separation. well explained examples about mentioned way are wave equation, heat equation, diffusion....
wave equation is Utt=C^2 .Uxx
in other word; derivatives of displacement to time, equals to derivatives of displacement to spatial multiplied by constant or vice versa.
however my equation is not like that and derivatives are multiplied to each other.for example : Uxx=(1+Ux)*Utt
Im wondering how to solve this equation.
I will be thankful to hear any idea.
Dear Alireza Akbari looks like your equation is a nonlinear PDE, there are tables for those:
However I could not find yours, but don't worry, I tell you a trick we use in MHD.
1. You linearized it, i.e., you solve the PDE as a function of ei(k.r - omega t)
2. You get a complex polinom, but I don't see any parameters in your equation.
3. Anyway you can try an algebraic manipulator such as math or maple and find the roots. However, I find it strange that there is not a parameter, you need it to scan the complex solution.
Best Regards.
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Hi everyone!
I want to optimize parameters (K, C, rho) in convection-diffusion equation for heat. I want to minimize the difference between the final simulation temperature and actual process temperature by changing the parameters in PDE equation. Please let me know about MATLAB tools and if there is any solved example etc.
Your guidance and opinion will be highly appreciated.
Thanks and Regards
Have you tried the software Mathematica? They have a free version online.
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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?
PDE's can be solved with the software Mathematica. They have a free version online.
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Definitely, I ask myself whether it was useful to introduce fractional partial differential equations.
As based on fractional differential operators this class of equations also surfers from the same drawbacks as the one described in:
Fractional partial differential equations were introduced to model anomalous diffusion, i.e. phenomena that exhibits power law behaviours other than 1/2. But it was shown recently that these kinds of behaviours can also be obtained with classical partial differential equations with spatially variables coefficients:
Where it is difficult to propose a physical interpretation with the fractional partial differential equations, classical partial differential equations with spatially variables coefficients allow interpretations relating to the geometry of the systems studied.
Thank you for your reply. But It is not because fractionnal partial differential equations have been widely used to describe many physical phenomena that it was a good idea and that we cannot do otherwise, with a greater physical sense and without the drawbacks associated to fractional operators (physical inconsistence among others).
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aaaaa aaaa aaaa
Under what circumstances a source term is added to continuity, momentum and energy equations?
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I work on spectral method for solving ODE and PDE. If you have any equation to solve, send me the govening equation, the boundary conditions and the initial conditions and I can propose a matlab code to solve these equations.
Best regards.
Dear Slim，
I have an equation, as shown in the picture in the attachment. I want to know its eigenvalues. I did it with spectral methods in MATLAB, but I don't know if I'm right. Can you help me calculate this eigenvalue?
Thank you very much!
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Pleas see the attached file.
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Hello.
Any idea to choose a suitable journal for a manuscript, whose abstract is given below? More specifically, a high-quality journal that makes a final decision within three months or so.
Thank you so much.
In multiscale modeling of subsurface fluid flow in heterogeneous porous media, standard polynomial basis functions are replaced by multiscale basis functions. For instance, to produce such functions in the mixed Generalized Multiscale Finite Element Method (mixed GMsFEM), a number of Partial Differential Equations (PDEs) must be solved. Thus, it makes sense to replace PDE solvers with data-driven methods given their great capabilities and general acceptance in recent decades. The main purpose of the present study was to develop four distinct Convolutional Neural Network (CNN) models to predict four different multiscale basis functions for the mixed GMsFEM. These models were applied to 249,375 samples, with the permeability field as the only input. The statistical results indicate that the AMSGrad optimization algorithm with a coefficient of determination (R2) of 0.8434 - 0.9165 and Mean Squared Error (MSE) of 0.0078 - 0.0206 performs slightly better than Adam with an R2 of 0.8328 - 0.9049 and MSE of 0.0109 - 0.0261. Graphically, all models precisely follow the observed trend in each coarse block. This work could contribute to many domains, especially the determination of pressure, velocity, and saturation in the development of oil/gas fields. Looking at this work as an image (matrix)-to-image (matrix) regression problem, the constructed data-driven-based models may have applications beyond reservoir engineering, such as hydrogeology and rock mechanics.
Submit your research to a journal that has published research that matches your research
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I am new to MATLAB and want to solve the following equation (in image).
Here c1 and ck are functions of the variable z. v1, D1, vk, Dk, k, Kd and c_cmc are all constants.
Boundary conditions are-> c1(0)=0, c1(0.1)=0.025, ck(0)=0.125, ck(0.1)=0
Even with other values of constants, how do I solve such type of equations?
Hi Atharva. As Silvia Trevisan says, your system is a system of ordinary differential equations (ODE). Because of of second derviative, is a system of order. In this case, you can introduce the two auxiliar variables:
v=dc/dz and w=dck/dz and tranform the original system in a FIRST order ODE system, that you can solve with matlab using any ODE solver as explained here:
Best regards and good luck!
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In science and engineering, one often solves a perturbed model representing; an ODE, a PDE, an algebraic equation...
Are we deliberately perturbing the equations and solve or what is the physical meaning?
It may also occur during Lab based practical experiments or what?
How does it affect the model or equation?
So many thanks!
Hi Dr Tekle Gemechu . Simply it means modified mathematically for the correction .
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PDE is
(1/r)d/dr(rdT/dr) = C*exp(E*(1-(1/T))
Boundary conditions are: At r = 0; dT/dr = 0 and at r = R0 T = Tw
where C, E, and Tw are constants.
Hello Siddiqui
Best Wishes
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In 2010, Dr. Khmelnik has found the suitable method of resolving of the Navier-Stokes equations and published his results in a book. In 2021, already the sixth edition of his book was released that is attached to this question for downloading. Here it is worce to mention that the Clay Mathematics Institute has included this problem of resolving of the Navier-Stokes equations in the list of seven important millennium problems. Why the Navier-Stokes equations are very important?
I finally could check the PDF, Prof. Aleksey Anatolievich Zakharenko
Dr. Khmelnik uses a variational principle to solve the NS equation, which is very powerful indeed.
He also discusses and gives examples & a reason for turbulence.
I know that the solution of NS is a non-linear problem that involves several modes and that it depends on the source.
However, my knowledge of the foundations of NS is very limited to a few linear/non-linear problems on non-equilibrium gas dynamics& MHD solved by the method, Prof. Miguel Hernando Ibanez had.
Thank you for sharing the link. I recovered my account.
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Dear RG members, I come across an extension of Green's function technique to nonlinear differential operators that can treat inhomogeneous terms present in the nonlinear ODE cases from Ref. and few following works. But I am unable to find any extensions to treat the nonlinear PDE cases with inhomogeneous terms using green's function techniques. Are they developed? And where I can find more information. Kindly suggest if known.
You can convert PDE to ODE by using one of the methods such as variable separation, traveling wave, or self similarity and then continuing work in the same way as in ODE. Best regards
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Sometimes to solve a PDE, the PDE is converted into an ODE of abstract form and then the solution of the abstract form is said to be the solution of the given PDE. What are the methods available in the literature to convert a PDE into an abstract form ODE?
If what is meant by "abstract" relates to ODE on functions spaces, e.g., Hilbert or Banach, etc., then the theory had been around for quite awhile is one of the main drivers for the theory of semigroups of operators. The setting is an mapping from an interval of the real one where for each t, A(t) is an unbounded closed densely defined operator on the appropriate function space. That is A could be an operator associated with a PDE on the appropriate Sobolev space. The the equation x'=A(t)x is an ODE on this function space.
The classical heat and wave equations can be viewed as ODE on the appropriate Banach or Hilbert spaces and the research in this area has been quite active since the 1940's.
There are several issues that need to be address. One is often these problems are ill-posed. The second issue is establishing the regularity that insures that weak solutions lead to strong solutions. A good place to maybe start is the 1948 work of Einar Hille where he introduced semigroups of operators, to address the Cauchy problem in Banach spaces. This theory mimics much of the theory of ODE with the appropriate conditions on the mapping t-> A(t).
Much of the theory of ODE in Banach spaces mimic that for ODE on finite dimensional space, however, care has to be taken because unlike finite dimensional space, the operators A(t) are not bounded. The second issue is the methods involved involve weak solutions and special conditions are required to establish that the weak solutions are strong solutions with the appropriate regularity.
This approach was key to the Sampson-Eells theorem where Joe Sampson and Jim Wells proved the existence of harmonic mappings between manifolds under certain conditions by solving a non-linear Cauchy problem for a non-linear "heat equation", replacing boundary conditions with homotopy conditions and using regularity theorems from elliptic PDE to establish regularity of the weak solutions.
In a follow on paper, Philip Hartman showed that expanding some techniques from stability theory of ODE to the nonlinear heat equation used by Sampson and Eells, expanded the results for Sampson and Eells and removing some of their "technical" conditions used to obtain the result.
Some other general references.
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There are many references in math and physics that describe how to write and solve the Diffusion Equation in different coordinate systems when the spatial dimension (d) of the problem is an integer, and ordinary diffusion is taking place (with no heavy tails).
If the spatial dimension of the region in which the diffusion is occurring is a non-integer,
and ordinary diffusion is taking place,
how do you write the Diffusion Equation for this type of problem,
and what are the methods for solving it for a specified initial condition and boundary conditions.
Any information about solving problems like this
or suggestions for references related to this topic will be greatly appreciated.
I have also solved the Diffusion Equation in arbitrary spatial dimension d for a ring concentration pulse that diffuses freely and is in between two circular absorbing boundaries which are each at a particular radius.
The solution to this problem can be found in the paper:
R. Zamir, "Finding the Green's Function, Concentration, and Survival Probability for the Finite Interval Diffusion Equation Boundary Value Problem in arbitrary spatial dimension d", which is available on ResearchGate.
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Either an elliptical plate solicits in biaxial traction by 4 concentric nodes, only the surface forces that the Matlab Pdetool software accepts in the boundary conditions window, how can I apply a concentric force.
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Hello, I went through the Fluent Theory Guide and other Ansys learning resources, however I couldn't find the answers to the following issues.
Let's discuss the problem on the example of a general steady RANS k-omega case.
No slip Wall
U is set to zero, k is set to zero, omega is given by a specific equation.
1. What about pressure? Does Fluent use the zeroGradient condition for pressure on a wall? It's done by default in OpenFOAM.
2. NS momentum equations are 2nd order in space, so the boundary conditions should involve the normal derivative (e.g. dUn/dn=0), if I am not mistaken ?
3. Consequently, the turbulence transport equations are 2nd order in space as well, so should their respective derivatives be specified on a wall?
Velocity Inlet
U, k and omega are specified.
5. Refering to question 2,3 - should a Neumann BC be specified for U, k, omega?
Pressure outlet
Fluent Theory guide says that all variables (apart from pressure) are extrapolated from the interior. However, that does not sound right. Imagine a highly diffusive system, wherein turbulence (or any other scalar) is diffused upstream, faster than the downstream convection. In such a case, turbulence properties at the outlet would affect the solution in the domain.
6. As a consequence, what BC are imposed for U, k, omega? Mentzer suggests "intervals" which freestream turbulence properties should be taken from. https://turbmodels.larc.nasa.gov/sst.html
Symmetry
Fluent specifies zero convective/diffusive flux across the symmetry plane.
7. We should specify a value of k and omega at the symmetry anyway... right? It has to be done in OpenFOAM...
Thank you very much for your time and effort, feel free to make a comment on that.
The boundary conditions depend on the case by case basis. It also depends on whether you are doing for incompressible flows or compressible.
Suppose you are working with incompressible flows and you have no idea about the initial pressure condition in that case you can opt for velocity inlets. In the same way if you don't have any idea about the velocity at inlet then opt for pressure inlets or mass flow inlet. If you use pressure inlet then always try to provide pressure outlet condition at the outlet. The distance between the inlet and the outlet should be enough to monitor the development of the flow. If the flow is not developed then the boundary conditions will not work for you so the geometry is also important to be understood. In such cases where you assume the the distance between inlet and outlet is very large but you provide a definite small value then try using Zero gradient boundary condition at the outlet to let the system extrapolate values from the region inside the flow domain.
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Could someone provide the description/representation of nonlinear subdiffusion equation?
I also like to know the physical significance and the difference between diffusive and subdiffusive PDE.
Amitha Rao
Non linear parabolic PDE -
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I am looking for the Matlab code to solve PDE using Deep neural networks. If anyone could help me. Thanks in advance.
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In Ansys dynamics explicit module, I have modeled a water tank for an earthquake analysis to observe sloshing behavior. I am interested in how Ansys solves the problem using conservation of mass, momentum, and Energy. I have used the eulerian body for water and the lagrangian domain for the water tank. My question Is it using the Finite Element Method (FEM) or Finite Volume Method (FVM) or Finite Difference Method (FDM)? to solve the PDE's
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How to find the exact solution of this PDE?
give me roughly idea or method please.
thank you.
https://www.researchgate.net/profile/Dmitry-Kovriguine-2 - In the query whether hyperbolic is specified ? . The ref gives all the probable methods of solutions of PDE . Although I am not expertised in the specific field but linear hyperbolic field is specified here -http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc2.htm
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Question about the loss of hyperbolicity in nonlinear PDE: when complex eigenvalues appear, what is the effect on flow? I understand that we do not have general results on existence in this case, but is it only the mathematical tools that are lacking where can we show physical phenomena of instability?
Blow -up -analysis -
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VOF is generally used for interface reconstruction. However, VOF discretization scheme with a special interface treatment is used to reduce the numerical diffusion and maintain the interface sharpness. Does the VOF discretization scheme use to discretize PDEs?
Nichols, B.D. and Hirt, C.W., “Methods for Calculating Multi-Dimensional, Transient Free Surface Flows Past Bodies,” Proc. First Intern. Conf. Num. Ship Hydrodynamics, Gaithersburg, ML, Oct. 20-23, 1975
1975
Hirt, C.W. and Nichols, B.D., “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries,” Journal of Computational Physics 39, 201, 1981
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I'm solving an eigenvalue problem in COMSOL. In the coefficient-form PDE interface, the coefficients of my partial differential equation contain eigenvalue terms (λ). However, if I enter "lambda" as the eigenvalue in the coefficient field as described in the COMSOL tutorial (https://doc.comsol.com/5.5/doc/com.comsol.help.comsol/comsol_ref_equationbased.23.011.html), the software will show it as an undefined variable. Then I can't solve the problem.
So, What is the expression for the eigenvalues in coefficient form PDE interface (eigenvalue study) in COMSOL? How can I enter the coefficients with eigenvalue terms (for example, as shown in the picture)? Thanks very much for your help.
It doesn't work either.
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I am using coefficient form PDE and the coefficients in the equation are in the matrix form(e.g. ea, da, c, etc). However, I don't konw how to enter a matrix in the coefficient form PDE interface. The matrixs are undefined variables that can't be used in this interface.
So how can I enter a matrix variable as a coefficient in the coefficient form PDE interface? Thanks for your help!
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I need the basics for including energy loss in Hamilton formulation for Finite element analysis for vibration of viscoelastic materials. The papers I read use complex modulus to represent viscoelastic losses or convolution integrals. Can someone give me a link where the formulation starts from Hamilton's principle?
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A lot of time i found different papers in which numerical methods execute to find the solution of given model these numerical methods apply with initial condition but in analytical methods we used wave transformation to convert PDE into ODE. Here my question is raised in any case there is numerical method in applied and wave transformation is not given and we want to apply the analytical method, analytical method need wave transformation so, how can we make a wave transformation?
If you are on a d-dimensional interval, you can expand the solution in the Fourier basis, then rewrite the PDE in this basis. For example, the heat equation would turn into a system of decoupled linear ODEs, with coefficient given by the eigenvalues of the Laplacian. This idea is the basis for the so-called spectral methods (see for example the book by Canuto, Hussaini, Quarteroni and Zhang).
Numerically, the initial condition is transformed into the Fourier domain by Fast Fourier Transform (FFT), and the final solution is transformed back into the original domain by Inverse FFT.
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I'm looking for a way to solve the diffusion equation with a non-linear source term, Equation 1 in the attached document, using the Crank-Nicolson method.
Using an implicit time integration method leads to a non-linear problem which has been solved at each time step as the one given in the attached document. For this purpose, a linearization is usually considered as Pr Filippo Maria Denaro mentioned in his answer. For example, Newton-like iteration approaches can be used when a second-order time accurate scheme is considered for the time integration.
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One of the first things we learned about PDEs, is how the cylindrical harmonics (e.g. the Bessel functions J, I, K, and Y ) can be used to solve cylindrical boundary conditions. However I have never seen a paper discussing elliptical boundary conditions.
1) Is there another function that can be used for elliptical boundary conditions (NOT an ellipsoid but a prism with an elliptical cross-section) ?
2) What is the relation of these elliptical harmonics to the cylindrical harmonics?
3) If the answer to the above is yes, how might I compute these efficiently in MATLAB?
4) Is there a corresponding analog to Graf's addition theorem within the open literature?
Best Regards and Thanks!
see
Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries
Michael F. Modest, Jun Yang
Journal:
Journal of Quantitative Spectroscopy and Radiative Transfer
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Hi there. Im writing a program in C to solve the viscous burgers equation in 1D using the finite volume method. I am currently using a structured uniform cartesian grid as my computational grid. However i want to implement adaptive mesh refinement in my program. Could anyone tell me the steps i should take in order to do this. Would i have to make i have to make an additional function like say Adapt_Mesh() as well as my Init_Mesh() function and call the Adapt_Mesh() function in the main time loop every time the mesh needs refinement and how could i go about determining if my mesh needs refinement or how would i go about this. Im relatively new to the realm of CFD hence im struggling a lot by thinking how to go implementing this. Could anyone help me. Id be willing to share my current code if anyone could point me in the right direction in regards with how to get started.
*EDIT: Also i have a current gird with 100 cells. If was to adapt the mesh would it refine the mesh so that NX is always 100 ie. by making areas of less density and areas of higher density so that the total points add up to 100 or does the adaptive mesh increase the number of total cells everytime the mesh is refined. This is something im unsure of.
If anyone is curiosus i have attached my program below just incase anyone wanted to take a look
If you want to increase your definition step by step, you can add a parameter to your array or similar data structure:
-step 1 a[i,j], that's your current description with large cells
-step 2 a[i,j,k,l], that's the higher resolution with smaller cells in each of the previous large cells
within the neighborhood of each (i,j), you introduce further divisions (smaller cells: parameter k horizontally and l vertically)
etc...
In this way you can always go back to less definition.
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If anyone could help me. It is my dissertation work. Thank you. I am looking for the matlab code to solve PDE using RBF.
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(Note: Problem Solved)
I am trying to define the phase field equation in the picture, using Coefficient form (or maybe convection-diffusion) PDE in COMSOL. However, I have a problem defining the second line, as there is no equivalent for these two terms in the COMSOL pre-defined equation. How would you do that?
#COMSOL #PDE #Phase-Field #PF #phasefield
You can find it in coefficient form PDE module. Please note that, you should define the units of your parameters.
You can, also, use the pdf form manuals which provided by software.
Regards,
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I am dealing with quasi-dynamic seismic cycle simulation, which involves 4 PDEs (or ODE). According to the Buckingham Pi theorem, there are many non-dimensional numbers in this system. How can I extract meaningful ones from such a PDE system? All examples in textbooks only consider one equation.
Another way I think may work is just to undimensionalize each of the equation in the equation system by first doing a scaling to the variables that change with time. We have four variables here, so four yet undetermined characteristic scales, which can be used to simplify the dimensionless equations (we can let four coefficients to be 1, for example). After doing this, we can get a dimensionless equation system, with the coefficients in front of each term as non-dimensional numbers that we want to find.
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I solved a nonlinear PDE by linearization and using finite difference and spectral methods. the errors for simple u_exact= exp(-t)(-8x^2-8y^2+7/2) don't change when the meshes become finer, in other word the error for all spatial meshes (20, 40, 60,...) is equal approximately, but for a more complex example u_exact=exp(-t)sin(x)cos(y) the results are very good and order of convergence is correct(i.e. order=2).
can anyone pleas tell me the reason of this trouble that happens in test whit a simple u_exacts?
I understood the reason of this trouble. there was a tiny mistake in my Implementation.
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Would prefer a book for learners.
see
Anh C.T., Hung P.Q., Ke T.D., Phong T.T.: Global attractor for a semilinear parabolic equation involving Grushin operatot. Electron. J. Differ. Equ. 32, 1–11 (2008)
D’Ambrosio L.: Hardy inequalities related to Grushin type operators. Proc. Am. Math. Soc. 132, 725–734 (2004)
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I am looking for an analytical solution of the 2nd order PDE as in the title. The problem would seem simple: I am referring to the case of the desorption of a volatile component from a solution layer in a closed vessel of finite geometry (p, T = normal).
I just try to find the time to establish a gas-liquid equilibrium. I mean vertical 1D model with no horizontal gradients. Unfortunately the literature research was ineffective. The only suggestion I found in a certain monograph does not make any sense. (Concentration disappears when time goes to infinity).
I'd be greateful for any suggestions.
Hello.
Your intuition is quite right :-).
Second Fick law is a parabolic second order partial differential equation (PDE). It can be solved by technique called separation of variables on finite domain. One have to suppose that solution C(x,t) consists of two independent functions X(x) and T(t). Therefore ∂C/∂x=T∂X/∂x and ∂C/∂t= X∂T/∂t. Using this assumption the second Fick equation splits in two ordinary differential equations (ODE) for T and X.
Solution then can be found in form of infinite Fourier series, therefore you have had the right feeling about it.
BUT, existence and uniqueness of PDE solution is strongly dependent on boundary conditions and initial condition. Although you already suggested the boundary conditions of Neumann type, the initial condition in form C(x,0)=f(x) is still required
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First of all I should say that I work in probability and my knowledge about PDEs is quite small, so this question could make little sense, please let me know if something is not well stated.
While dealing with approximations methods for SDEs a I've noticed a particular connection between an SDE and a deterministic PDE of the form:
u_t+σ(t)⋅ u_x+x⋅u=b(t,x,u)
where b is a Lipschitz continuous function in u.
I've tried searching online for application of this kind of equation but unfortunately I wasn't able to find anything concrete. In the book by Moussiaux, Zaitsev and Polyanin "Handbook of first order PDEs" they discuss methods for solving this kind of equations but they don't provide examples of applications.
I suspect this could be connected somehow to the transport equations, but I am not entirely sure. Do you know of some references for applications of this particular equation?
Filippo Maria Denaro Grazie per la tua risposta, dovrò cercare un po' circa la equazione di onda con coefficienti variabili. La verità è che io lavoro in calcolo stocastico da una prospettiva matematica, questa equazione è venuta fuori e volevo capire se avesse una qualche rilevanza dalla modelistica.
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Hi,
I build a model which includes 2 parts, each has a specific constant material property. For example, part/domain 1 has the absorptions coefficient a=50 and the other part/domain of the model will be a=100. Now I use the general coefficient form PDE and want to assign those absorptions coefficients in the PDE dialog. How can I do this?
Thank You, Chironjeet
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UPD (29/08/2018): Is the RANS/LES way of turbulence modeling a converging process, i.e. will it hypothetically converge to a hypothetical big but single PDE system? Or will it inevitably disassemble into several non-overlapping PDE systems/approaches each of which is valid only for a certain turbulent regime?
After more than 5 decades of active search, it seems that the hierarchy of moment equations for the Reynolds stress and higher moments derived from the Navier-Stokes equations does not have a universal closure that could be applied to more or less wide range of flows. Numerous closures have been proposed since then. All of them contain heuristic arguments which are usually changed from case to case. So, does it mean that our understanding of turbulence will never be complete? Or do people still believe that a “magic closure” exists which can explain everything?
Completely agree with you in the computational context, not for the real-world scenarios. What I mean is somehow deal with the butterfly effect which implies even the farthest smallest decimals may have a role to play in the holistic simulation of turbulence like in nature.
Regards,
Hamed
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hello
I need to solve a set of the nondimensional equation in comsol (PDE module). for this reason, I need to change the gradient operator(in z-direction I need to add coefficient).how can I do this?
thanks.
Galerkin method.
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I want to find the analytical solution of a radial diffusion system (1D cylindrical coordinates). Here what is considered is pressure diffusion. The conditions are simple and similar to the following.
Initial P = P0
P(r=infinite) = P0
P (r= 0) = P1 when t > 0
I need the solution of P at any r_i and t.
The flow rate term is unknown.
Could anyone give any suggestion? Thank you.
see
Radial symmetry of solutions to diffusion equations with discontinuous nonlinearities
Joaquim SerraJournal:Journal of Differential EquationsYear:2013
best luck
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Dear colleagues,
I would like to know how to implement the elasticity equation for an isotropic solid material ( elasticity equation , attached) using the PDE weak form interface in COMSOL Multiphysics. The PDE weak form requires three dependent variables [u, v, w] to represent the displacement vector U. Therefore three weak expressions must be defined in comsol instead of the general weak expression for the elasticity equation (weak form attached, where u is the displacement vector and w is the test function).
How can I replace the general weak form expression(weak form attached) into three weak expressions that would be implemented on COMSOL.
Many thanks.
Please find the attached files. Let me know if you have any questions.
Thanks and best
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Hello all,
I am looking for an method / algorithm/ or logic which can help to figure out numerically whether the function is differentiable at a given point.
To give a more clear perspective, let's say while solving a fluid flow problem using CFD, I obtain some scalar field along some line with graph similar to y = |x|, ( assume x axis to be the line along which scalar field is drawn and origin is grid point, say P)
So I know that at grid point P, the function is not differentiable. But how can I check it using numeric. I thought of using directional derivative but couldn't get along which direction to compare ( the line given in example is just for explaining).
Ideally when surrounded by 8 grid points , i may be differentiable along certain direction and may not be along other. Any suggestions?
Thanks
The answer to a question about the numerical algorithms for resolving the issue of differentiability of a function is typically provided by the textbooks on experimental mathematics.
I recommend in particular: Chapter 5: “Exploring Strange Functions on the Computer” in the book: “Experimental Mathematic in Action”.
You can also get a copy of the text in a form of a preprint from
Judging by the quote placed in the beginning of Chapter 5, the issue of investigation of the “strange functions” was equally challenging i 1850s as it is 170 years later:
“It appears to me that the Metaphysics of Weierstrass’s function
still hides many riddles and I cannot help thinking that enter-
ing deeper into the matter will finally lead us to a limit of our
intellect, similar to the bound drawn by the concepts of force
and matter in Mechanics. These functions seem to me, to say
it briefly, to impose separations, not, like the rational numbers”
(Paul du Bois-Reymond, [129], 1875)
The situation described in your question is even more complicated because the function is represented only by a few values on a rectangular grid and it is additionally assumed that the function is not differentiable at a certain point. In this situation I can suggest to use the techniques employed in the theory of generalized functions (distributions).
For a very practical example you can consult a blog: “How to differentiate a non-differentiable function”:
In order to answer your question completely I would like to know what is the equation, boundary conditions and the numerical scheme used to obtain a set of the grid point values mentioned in the question.
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For example: Lu+0.5u=f(x,y)
or Lu-au=o (elliptic PDE).
Actually I am looking some PDE for which I should get a singular system after applying any finite difference scheme.
Dear;
For produce singular system you use eigenvalue and with Dirichlet’s or Neumann’s boundary conditions will .
Regards
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