Partial differential equations (PDEs) are equations that relate the partial derivatives of a function of more than one variable. This tag is intended for questions on modeling phenomena with PDEs, solving PDEs, and other related aspects.

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5
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0answers
62 views

Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \begin{equation} \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - ...
1
vote
1answer
36 views

FreeFem user-defined function

I'm trying to solve the steady-state heat equation with a space-dependent thermal diffusivity in FreeFem. I.e. $$\nabla\cdot(\kappa(x,y)\nabla T) = 0$$ I'm hoping to read in from a file locations ...
5
votes
0answers
23 views

Phase dislocations and numerical accuracy

I am solving the nonlinear Schrodinger equation (NLSE), $$A_t+iA_{xx}+i|A|^2A=0$$ where $A$ is a complex valued function, which can be written as $A=ae^{i\theta}$ for $a,\theta$ real. Now, for ...
4
votes
1answer
116 views

High order unconditionally stable discretization for a scalar hyperbolic PDE

In order to numerically solve the following differential equation: \begin{equation} \text{Fr}\{f\} := v(k)\dfrac{\partial f(z,k)}{\partial z} - F(z) \dfrac{\partial f(z,k)}{\partial k} = ...
1
vote
2answers
57 views

The second variation of displacement interpolation function in Finite Element Method

I need to calculate the second variation of displacement interpolation function $u = \sum N_a u_a$ in Finite Element Analysis, where $N_a$ are the shape functions and $u_a$ are the nodal values. ...
4
votes
1answer
62 views

What is the preferred method for evolving the Nonlinear Schrödinger Equation?

I am interested in evolving the (cubic) self-focusing nonlinear Schrödinger equation, $$i\frac{\partial \psi}{\partial t}+\frac{1}{2}\frac{\partial^2 \psi}{\partial x^2}+\left|\psi\right|^2\psi=0$$ ...
3
votes
1answer
120 views

Is this finite difference approach correct?

I am solving incompressible 2D Navier-Stokes equations with zero y-component velocity. The geometry is a simple 2D pipe of a length $L$ and diameter $W$ and there is only two boundary conditions: ...
0
votes
0answers
73 views

How to solve a set of 1st order 1D PDEs in Matlab?

i have already asked something similar to this once but sadly i wasn't able to get a complete answer, so i remain stuck on this problem. the Matlab function pdepe seems to only be able to solve 2nd ...
0
votes
1answer
31 views

How to use time delays in the solvepde function in MATLAB for a system of PDEs?

The solvepde function was introduced in MATLAB R2016a. I am able to solve my system of PDEs if there are no time delays involved. Does anyone know how to include time delays in the solvepde function?
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0answers
20 views

Convergence of the stress at an interface using AMR with quadtree meshes in a solid mechanics problem

Assuming a solid mechanics problem, linear elasticity, with a domain split in two by an interface that is not aligned with the mesh. The mesh is a quadtree with squares. There are different material ...
0
votes
1answer
155 views

How to solve my PDE's in Matlab?

I need to solve a set of 5 PDEs for functions $u(x,t)$. I looked up the Matlab function pdepe. It looked perfect for my case, until I read the line: ...
1
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0answers
60 views

Stability of two PDEs

I have the following two PDEs which I want to check for stability: $$u_t= u_{xx} , \ u(x,0)=1 , \ u_x(1,t)=-hu^4(1,t) , \ u_x(0,t) = 0 $$ $$u_t = u_{xx}-\sin(x+t)+\cos(x+t) , \ u(x,0)=\cos(x) ,\ ...
2
votes
1answer
115 views

How do I program periodic boundary conditions? [duplicate]

Hi I have a code below that solves non linear coupled PDE's given Dirichlet boundary conditions. However I need to implement periodic boundary conditions. The periodic boundary conditions are ...
-1
votes
1answer
75 views

Stability analysis for coupled nonlinear system of partial differential equations

I'm trying to solve a nonlinear partial differential equation \begin{equation} L(u_{xxtt},u_{xx}u_{tt},u_{xt}^2,u_{xt},u_{tt})=0 \end{equation} using finite difference methods. In order to remove the ...
6
votes
1answer
156 views

What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the ...
1
vote
3answers
202 views

Algorithm suggestion for PDE - example: heat equation

I want to solve the PDE equation numerically. For this, I started my study with something simple; heat equation $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial^2 x} $$ with the initial ...
4
votes
1answer
56 views

Boltzmann equation and equilibrium distribution function

The free-streaming term in stationary 1D Boltzmann equation satisfies the equilibrium distribution function, that is: \begin{equation} \text{Fr}\{f\} := v(k)\dfrac{\partial f(x,k)}{\partial x} - ...
3
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0answers
39 views

Convergence of KKT equations for discrete parameter estimation problems

Consider a discrete constrained optimization problem: $$ \mathbf{q}_*^h= \arg \min {\cal J}^h(\mathbf{x}^h[\mathbf{q}^h],\mathbf{q}^h) $$ subject to the (weak-form) constraint $$ ...
0
votes
3answers
200 views

Numerical simulation of a reaction-diffusion system on MATLAB with finite difference discretization of spatial derivative

I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form: ...
3
votes
0answers
91 views

Symplectic integration of PDE

I consider ordinary wave equation $$ u_{tt} - u_{xx} = 0 $$ with initial conditions $$u(x, 0) = \exp (-2 x^2) \\ u_t(x, 0) = 0 $$ To solve this problem I approximate $u_{xx}$ with 4-th order ...
1
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0answers
57 views

Solving system of equations with zeros on diagonal [closed]

I am using finite elements, Newton Raphson technique and MUMPS direct solver, to solve for $P$ in this equation: $$\frac{\delta K}{\delta x} \frac{\delta P}{\delta x}= \frac{\delta K B}{\delta x} + ...
1
vote
0answers
51 views

PDE discretization (via finite difference sheme) question

So after posting this question and reading all your comments I would like to make this new question (update). If you consider the three equations presented here: $$\frac{\partial \rho}{\partial t} ...
1
vote
1answer
54 views

Non-uniform finite difference Adaptive Mesh Refinement

Assuming that the crosses in the figure below are unknowns in a vertex-centered finite difference scheme in an Adaptive Mesh, how can I calculate the double derivate (Laplacian) at the Red x ? The Red ...
-1
votes
1answer
107 views

Solving coupled PDE in COMSOL [closed]

I have the system of equations \begin{align} &A \frac{\partial u_1}{\partial t} = 1 - u_1 B \frac{\partial u_2}{\partial y}\\ &\frac{\partial u_2}{\partial t} = \frac{\partial}{\partial ...
0
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0answers
38 views

Entering an Electric field dependent equation in material contents in COMSOL?

I am trying to model a CNTFET in COMSOL. I am following the same steps that were used to model a MOSFET (provided in the semiconductor module manual), except for a few additional layers and materials. ...
1
vote
1answer
73 views

How to deal with PDE over the real line

I have a PDE defined over $\mathbb{R}$, for which I don't have the exact solution, and I am to approximate it with finite differences so I need to input some BC. Can anyone suggest any good ...
0
votes
0answers
38 views

Error norm in maple

I have the function $cos(x+t)$; the following PDEs have $cos(x+t)$ as the analytical solution of them: $$u_t = u_{xx} -\sin(x+t) + \cos(x+t) \\ u_x(0,t)=-\sin(t) \\ ...
6
votes
1answer
144 views

A simple PDE solution question

I need to ask a question about partial derivatives. I want to solve this equation (steady state, one dimensional continuity equation): $$\frac{\partial (\rho u)}{\partial z}=0$$ which is equivalent ...
4
votes
1answer
78 views

Refinement in AMR

Assume I start with an 8x8 coarse mesh (see Fig 1) where the vertices (except boundary vertices) represent the unknown variable. After iterative approximation - ...
1
vote
2answers
67 views

Fourier techniques and periodic boundary conditions

Could somebody explain to me why periodic boundary conditions are automatically satisfied if you solve your problem assuming a Fourier series? So, if we assume a Fourier series for our solution, we ...
0
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0answers
50 views

Simulation of parameter propagation of a numerical PDE solution

I have the following PDE: $$u_{xx}u^3 - \sin(xt)u_{tt}=u(x,t)$$ $$ u(x,0) = h,\ u_t(x,0) = 1-h,\ u_x(0,t)= h*u^3(0,t),\ u_x(1,t)= 1$$ where the problem is defined for $t>0$ and $x \in [0,1]$, and ...
7
votes
1answer
122 views

Solving a simple Schroedinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Piaevskii equation I have found a problem that can be tracked down to something buggy occuring in the simplest Schrodinger equation possible: $\partial_t \psi ...
3
votes
1answer
79 views

Comments needed on the doubts of PDEs in moving boundary problems

We know that in classical two-phase Stefan problems, let's say in the temperature distribution of ice-water problem here, the governign PDEs are: \begin{equation} \left. \begin{aligned} ...
3
votes
2answers
140 views

Are the Finite Difference and Finite Volume Methods different after the application of the Gauss Divergence theorem on the FVM?

I have a rudimentary question about the differences between finite difference (FDM) and finite volume methods (FVM). In FDM we concentrate on the nodes (points) in space while in FVM we concentrate ...
0
votes
0answers
86 views

Sobolev space H^2

In the definition of $H^2$ what does mean that the first and the second partial derivatives of some $u$ belonging to this space belong to $L^2$ in the meaning of distributions?
0
votes
1answer
81 views

Convection diffusion reaction equation (stiffness, solver)

I am trying to solve the CDR-Equation in 2D: $$\frac{\partial c(x,y)}{\partial t} + \nabla \cdot ( -d\nabla c(x,y) + \vec{v}(x,y) c(x,y))+ a c(x,y)=0\,,$$ with Boundary Conditions (length of square ...
0
votes
1answer
52 views

Looking for a matlab/maple code for plotting the truncation error

On page 18 on this text: http://www.dima.uniroma1.it/users/lsa_adn/MATERIALE/FDheat.pdf , the graph in figure 8 on this page, how would I write a suitable code in matlab or maple that will produce ...
8
votes
1answer
237 views

Strong vs. weak solutions of PDEs

The strong form of a PDE requires that the unknown solution belongs in $H^2$. But the weak form requires only that the unknown solution belongs in $H^1$. How do you reconcile this?
1
vote
1answer
68 views

Implementation of Neumann boundary condition with method of lines - 1D diffusion/reaction equation

I am solving the monodimensional diffusion/reaction equation by discretization using the well-known method of lines ${\partial c\over\partial t}=D{\partial^2 c\over\partial x^2}+ r\text{ for ...
-3
votes
1answer
142 views

How can I solve stiff equations by method of line (MOL)?

I want to solve 7 coupled equations.I use method of line(MOL) and discrete the equation in Length and radius and convert them to a system of ODEs in time.and use ode15s to solve them in MATLAB. But an ...
0
votes
0answers
45 views

Solving a nonlinear problem with CDF

I'm trying to solve this problem: $\begin{cases} \partial_t E=-k\left([f(\rho)-i.\left[\delta+\frac{1}{2}a\left[\dfrac{\nabla^2_{\bot}}{4}+1-\rho^2\right]\right]]E - 2CP\right)\\ \partial_t ...
5
votes
1answer
131 views

PDE - Conservative form, conservative methods and discrete conservation

I cannot find a reference explaining clearly and rigorously the links between the notions of conservative form for a PDE, a conservative numerical method and discrete conservation. I would be very ...
3
votes
2answers
123 views

Solving coupled PDEs numerically on a semi-infinite domain with no-flux boundary conditions

I have the following system of PDEs for which I have given parameters $\gamma, \tau$ and $\mu$, $$\begin{align} T_t = &\ \gamma\,(L +\tau F-T)\\ F_t = & -F_x-(F-LT)\\ L_t = &\ \mu ...
2
votes
0answers
99 views

System of PDEs in Julia [closed]

I'm having difficulty coding and was hoping someone could see my error. I'm using Julia with the ApproxFun package and am trying to solve the wave equation with Dirichlet boundary conditions ...
0
votes
0answers
74 views

Solving Laplace equation on complex geometry using mixed boundary conditions

Consider a regular, Cartesian grid. The domain where I want to solve Laplace equation consists of a number of grid points on the grid. They form a complex geometry. I want to solve the Laplace ...
1
vote
0answers
18 views

Duality/Lagrangian condition and Variational Inequality of a cost functional

Given the functional $J_A(y,u) = \frac{1}{2}||y-y_d||^2_{L^2(w,\omega)} + \frac{\lambda}{2}||u||^2_{L^2(w^{-1}, \omega)}$ where w is a function belongs to Muckenhoupt class. Given the optimization ...
0
votes
0answers
47 views

Nitsche' method for handling Dirichlet boundary conditions [duplicate]

I want some explanations about the genesis of Nitsche' method Thanks
1
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0answers
81 views

Boundary Conditions for the given PDE

I'm working on the Black-Scholes equation, but I'm pretty new to financial modeling. Right now, I am trying to understand the Black-Scholes PDE. I understand that the Black-Scholes equation is given ...
0
votes
1answer
138 views

Heat Equation - PDE

I'm trying to model the Black-Scholes Equation (transformed into a heat equation) using method of lines in Python. The transformed formula is basically \begin{equation*} \frac{\partial u}{\partial ...
2
votes
0answers
76 views

Modeling First Order Parabolic PDE (Battery Storage Model)

I'm trying to solve the following first order parabolic partial differential equation, \begin{equation*} X \frac{\partial V}{ \partial Q} = -\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial X^2} + ...