Questions tagged [pde]
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.
883
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Solving 1D convection using method of lines
I'm interested in solving the following 1D-advection equation using method of lines.
$$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$
The spatial domain has been discretized into ...
- 421
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1
answer
107
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Using Kutta Merson on NLS
I'm trying to use the Kutta-Merson to get the same results as in the book Solitons, Nonlinear Evolution Equations and Inverse Scattering - M. J. Ablowitz - pg 140
The author propose using the Kutta-...
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1
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208
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Mixed formulation in 1D
I have been working on a hybrid dimensional model using the mixed FEM formulation, in which 3D elements and 2D elements are combined by certain relationships between the degrees of freedom (DOFs) ...
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2
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132
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Method to find PDE equation coefficient satisfying mean solution?
What is the best approach to go about solving a PDE problem of the type
\begin{equation}
k^3\Delta u - k(\mathbf{1}\cdot\nabla u) = 0\, ,\\
u=g\; \text{on}\; \Gamma_D\, ,\\
mean(u) = u_\text{...
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1
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285
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Libraries to deal with unstructured grids
I am dealing with a *.cgns file. This mesh format, when saved as an unstructured grid, holds nodes coordinates, nodes connectivity per element and boundary ...
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1
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116
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Multi-steps method for Navier-stokes equations with strongly nonlinear diffusion
I am trying to solve a particular form of the Euler / Navier-Stokes equations in 1D, with very strong and non-linear diffusion coefficients.
My system of equations is
\begin{cases}
\...
- 45
1
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1
answer
90
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Physical data for heat equation
I want to implement an algorithm to solve a heat equation, i.e.
\begin{align*}
\partial_t u - \Delta u = f \text{ in } \Omega\times(0,T)\\
\partial_nu = 0 \text{ in } \partial\Omega \times (0,T)\\
u(0)...
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2
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701
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Mass conservation in 1d diffusion by method of lines
I am solving the 1D diffusion equation by discretization using the method of lines. My problem is that I don't manage to ensure mass conservation. I have read many similar questions about the topic ...
- 215
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1
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Proper boundary conditions for potential flow around cylinder
I am computing the stationary, incompressible, inviscid and irrotational flow around a circular cylinder using a discretization in general coordinates.
I derived a PDE and proper boundary conditions ...
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1
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How to implement finite difference method for one dimensional Navier-Stokes PDEs
I am trying to use backwards finite difference method to numerically solve a pair of partial differential equations:
$\frac{\partial \left(pv\right)}{\partial x}+\frac{\partial p}{\partial t}=0$
$\...
- 147
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1
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Examples of finding eigenfunctions of coupled DEs
I am looking for examples of numerically finding eigenvalues/eigenfunctions of coupled DEs. If anyone is able to point me towards any examples, preferably with code included, it would be much ...
- 147
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1
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212
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Implicit method for two coupled PDEs
I have two equations (coupled), with the variables $T_1$ and $T_2$ and the constant $T_0$, which are (when written unitless, i.e. without prefactors):
$$\partial_t T_1 = 1-T_1^3+T_1+\nabla\left(\frac{...
- 553
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2
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170
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Improving calculation algorithm for coupled PDEs
I have the following two PDEs:
$$\partial_zU=\nabla_r^2U+\varrho U$$
$$\partial_t\varrho=a\vert U\vert^4$$
with $a$ a constant and
$$dt=dz\cdot\frac{n}{c}$$ with $n$ the refractive index of a ...
- 553
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1
answer
324
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An example of mixed elliptic problem using lowest-order Raviart Thomas element
I try to solve the following mixed second order elliptic PDE in the domain $D=[0, 1]^2$
\begin{eqnarray*}
v+\nabla p=&0 \quad &\text{in} \quad D,\\
\text{div}(v)=&1/2 \quad &\...
- 11
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4
answers
372
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Mass transport in porous media
My goal is to numerically solve the convection-diffusion equation of the form:
$$\frac{\partial C}{\partial t} = \nabla (D \nabla C) - \nabla (v C)$$
$C$ is concentration and $D$ is the diffusivity ...
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1
answer
408
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Numerical solution of non-linear heat-diffusion PDE using the Crank-Nicolson Method
I am trying to solve numerically the following 1D EBM:
$C\frac{\partial T[x,t] }{\partial t} - \frac{\partial }{\partial x}\left ( D(1-x^2)\frac{\partial T[x,t] }{\partial x} \right ) + I[T] = S[x,t](...
- 19
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1
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368
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Solving first versus second order PDE
I am trying to numerically solve a PDE, and just had a question as to the validity of a certain approach. For example, given the PDE:
$$ \frac {\partial ^2 E}{\partial t^2} = - k\frac {\partial E}{\...
- 578
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1
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91
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Name of third-party Matlab packages for solving PDEs
I am interested in working with systems of PDEs in two spatial dimensions, as part of a mathematical modeling project. The systems of PDEs are coupled, and there is no convenient reduction to 1 ...
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1
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318
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FD implementation of Absorbing Boundary condition for acoustic wave
I am simulating acoustic wave equation in which absorbing boundary condition has to be applied. It is applied in two ways
Ist method is as mentioned in this paper.
Boundary condition at bottom is (...
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2
answers
123
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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. ...
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1
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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 y}\...
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2
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Heat Equation Solution in One Dimension (x, t)
We're currently solving the heat equation as a part of the PDE sequence in class.
We've been given the formula:$$T(i, n+1) = T(i,n)+\alpha \left [\frac{T(i+1,n)-2 T(i, n)+T(i-1,n)}{\Delta x^2} \right ...
- 311
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1
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How can I solve coupled equations by the method of line(MOL)?
I want to solve 3 coupled PDEs equations. They depend on time, radius and length. I used the method of lines (MOL) and converted them to a system of ODEs in time. Now I want to solve them using MATLAB....
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1
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344
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Computation of plane wave scattering on semi infinite plane
I have attempted to code up the simple math required to plot the total field set up by an incident plane wave on a semi-infinite flat plate which can be found here.
To summarise:
$$\phi_s(r,\theta ) ...
- 873
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1
answer
571
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Raviart-Thomas elements global definition and compact support
As per the suggestion by Christian in the comments here, as part of my continuing quest to understand the Raviart-Thomas (RT) elements I'd like to know how exactly the RT elements are defined globally,...
- 615
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1
answer
141
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A doubt in Multigrid V-cycle
Assume I have 3 levels of grids. Finest Grid = level 2, Coarser Grid = level 1, Coarsest Grid = level 0.
Relax $u$ on $Au = b$ at level 2 for 3 times.
Find residual $r2$ at level 2, then restrict to ...
- 467
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1
answer
2k
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Instability of pdepe in Matlab... boundary conditions?
here is a Matlab beginner banging his head on the wall...
I am trying to solve a system of partial differential equations in Matlab, with both derivatives in time and space domains. I am using the ...
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1
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2k
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Calculating Divergence in COMSOL
Is it computationally safe and accurate to use the following equation in COMSOL to compute the divergence of the vector quantity J (instead of using its general built-in equations that have $\nabla$ ...
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3
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1k
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Capacitance in freefem++
I would like to simulate a capacitor in 2d with freefem++. This is the code I used:
...
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1
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778
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Von Neumann Stability Analysis
I came across the following task recently:
Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
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0
answers
91
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How to implement boundary conditions for the Thomas algorithm
For my variable $U(t,x)$, I have implemented the thomas algorithm with $U_j^i$:
$$ a(x)U_{j-1}^{i+1}+ b(x)U_j^{i+1} + c(x)U_{j+1}^{i+1} = d(x)U_j^{i} $$
Then $\textbf{A}$ is a tridiagonal vector with ...
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0
answers
82
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Seeking open-source PDE Solver for inhomogeneous material properties
I'm currently in search of an open-source PDE solver (Finite Element Method is preferred) that can effectively handle the challenge of material properties coefficients associated with each element in ...
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0
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44
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Solution to the Liouville-Gibbs equation
What would be the approach to numerically solve for $\rho(x,t)$ the following equation with some initial conditions
$$\frac{\partial\rho}{\partial t}
+\sum_{i=1}^n\left(\frac{\partial(\rho g_i)}{\...
- 397
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0
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96
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Method of lines for a mixed PDE
I am trying to solve the following PDE using the method of lines to discretize space, and then solve it as system of ODEs at each point in space using ODE15s:
subject to
and initial condition $w(z,t=...
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0
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209
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Using solve_ivp for a PDE: how to handle multiple time-dependent variables?
I am trying to build a Python code that solves a set of coupled differential equations which will be spatially discretized by the method of lines advancing in time. I am planning to use ...
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0
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95
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Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation
I am currently coding a solution to the following PDE:
$\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
- 11
1
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0
answers
182
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Solving PDE on a non-uniform grid with Crank-Nicolson scheme
I am solving a 1D diffusion-type equation with the finite-difference Crank-Nicolson (CN) scheme, and I need to densify the spatial grid around the central point. One could change the spatial variable ...
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0
answers
81
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Linear PDE solution with constraints
Consider the following linear PDE:
$$\nabla_q V(q) - M_d(q)M^{-1}(q)\nabla_q V_d(q) = 0,$$
where $V(q)$ and $M(q)$ are known and $M_d(q)$ is a grey box function (e.x., $M_d(q)$ is fitted using a ...
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0
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118
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Discrete laplacian 9 point
I am trying to write a code for 9 point discrete laplacian. I would like to write a matrix and solve the linear system $AU=F$ with gradient conjugate method.
I wrote the matrix this way
...
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0
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185
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Solving second order coupled differential equations in python
as I have to design a reactor and therefore have to get its length x, I have to solve the following differential equations:
$$D_{eg}\tfrac{d^2A_g}{dx^2}-u_g\tfrac{dA_g}{dx} = k_la_b\left(\tfrac{A_g}{...
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0
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110
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Crank-Nicolson vs Spectral Methods for the TDSE
The time-dependent Schroedinger equation (TDSE) depends linearly on the system's initial state $\vert \psi(0) \rangle$, such that the solution can be generally written as
$$ \vert \psi(t) \rangle = \...
- 111
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0
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98
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Numerical methods for $u_t+c u_x= \frac{-c}{x}u$?
I am looking for possible numerical methods to solve the PDE
$$u_t+c u_x= \frac{-c}{x}u$$
I am particularly interested in a Finite elements method, although I am also curious if you can expose some ...
- 151
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0
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75
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Finding the weak form of a PDE with a tensor argument
I am trying to solve for the order parameter ($A$) in the Ginzburg Landau equations. I had asked on the math SE site but was recommended to ask here.
We are trying to solve the following equation, (...
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0
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68
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Preserving conservation properties across time-integration schemes
I am interested in deriving an accurate, implicit discretization of a nonlinear advection-diffusion equation
$$
\partial_t f = \frac{1}{v^2} \partial_v J(f,v) \equiv F(f,v) \tag{$\star$}
$$
with flux
...
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How to numerically solve PDE that governs the free vortex wake model?
Crossposted at Math SE
I am reading a paper on the free vortex wake model for a helicopter rotor blade, which is described by the following PDE:
$$\frac{\partial \vec{r}}{\partial \psi} (\psi, \zeta) ...
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0
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58
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Stability plot of upward difference implicit time
I am analyzing the stability of 1D convection (advection) equation as shown in the picture. When I derive the equations as shown I want to get rid of the complex number.
I`m asking if those stability ...
1
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0
answers
68
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discretization of advection diffusion with variable coefficients
I am looking for help to find a somewhat stable FD numerical scheme for the advection diffusion equation posed on a curve $(x(r),y(r))$. The equation becomes
$$u_t=\alpha(r) u_r +\beta(r) u_{rr}+f(r,t)...
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0
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139
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Solving Laplace equation with constraint on boundary
I have found the following PDE problem in a paper:
Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
- 227
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0
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158
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Integrating a wavelike equation with absorbing boundary conditions
I am trying to numerically solve the following equation:
$\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+V(x) \phi(x, t)=0$ On some domain, with:
$\phi(x, 0) = I(x)$
...
- 11
1
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0
answers
126
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ode15s command for coupled PDEs
I am trying to solve the coupled PDE system given here-Solve System of PDEs, using the method of lines and the ode15s command. Referring to variables $u1$ and $u2$ as $u$ and $v$ respectively, I have ...
