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# Questions tagged [differential-equations]

For questions about solving, analyzing, or creating differential equations to model some system. If possible, include specific tags about the type of differential equation (e.g. [tag:pde], [tag:ode], [tag:stochastic-ode]).

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1 vote
51 views

### Solving TOV equations that describes neutron stars in modified f(R, T) gravity

Sorry for the long post, tldr at bottom. I'm trying to use standard RK4 code in C/C++ to solve a coupled system of 2 modified TOV equations in f(R,T) gravity and reproduce some of the results of this ...
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344 views

### Time integration of first-order ODE with higher-order information

Suppose I wish to derive a numerical integrator for the first-order ODE $$x'(t)=F(x(t)).$$ By differentiating both sides of the expression in $t$, I can write a second-order relation also satisfied ...
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1 vote
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### Moving least square method in finite volume method

Consider a differntial equation like : $\nabla .(\nabla u)=cte$, using finite volume method we can write $\int\nabla .(\nabla u) dv =\int n.(\nabla u) dA$. Here we want to use the moving least square ...
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1 vote
202 views

### Numerically solving the Advection-diffusion equation with no-flux boundary condition leads to violation of mass conservation

I am trying to solve numerically the advection-diffusion equation of the following form $$\frac{\partial C}{\partial t}=\alpha\frac{\partial^2 C}{\partial x^2}+\beta \frac{\partial C}{\partial x}$$ ...
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135 views

### Galerkin Method - Why does integration-by-parts eliminate need to enforce Neumann boundaries?

I've posted this to MathStackexchange but I figured I'd also post here as well as I have yet to receive an answer on my original post, and that I would be more likely to encounter users of the ...
1 vote
112 views

### Can I combine the backward and forward euler methods - simialr to modified euler method?

Constructing Modified Euler Using the same strategy as done in the construction of Modified Euler. Starting from Trapezoidal Method $$y_1 = y_0 + \dfrac{h}{2}\left(f(x_0,y_0) + f(x_1,y_1)\right)$$ ...
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1 vote
122 views

### Is there any way to reduce an RK4 method's dependence on step size?

I am working in the sphere of orbital simulations, where orbital trajectories are computed using the differential equations describing gravity. Due to the great timescales of orbits, a step size of ...
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1 vote
71 views

### Why does a two-body simulation result in no change of the y-component?

I've been attempting to create a model of a heliocentric orbit based on Newton's law of gravitation: $$\frac{d^2 \vec r}{dt^2} = -\frac{GM}{|r|^2} \hat r$$ This is what I have so far: ...
• 133
951 views

### Numerical implementation of ODE differs largely from analytical solution

I am trying to solve the ODE of a free fall including air resistance. I therefore defined my ODE as: def f(v, g, k, m): return g - k/m * v**2 which in my ...
• 123
199 views

### Using Sundials CVODE in MATLAB

I'm currently using ode15s to solve a set of stiff differential equations. I am trying to use the MATLAB profiler to understand the section of the ode solver code which calls BLAS routines. Since the ...
• 433
669 views

### Is my differential equation solving code wrong?

I am trying to simulate LLG equation without damping. The equation is $$\frac{d\vec{m}}{dt} = \vec{m}\times\vec{H}$$ I am solving in spherical coordinates as LLG equation is known to have problems in ...
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1 vote
39 views

### How does the time needed for force propagation increase with the discretization (using symplectic Euler schemes)?

I am trying to model a physical system by (lets say the system is a long deformable object, on which I can apply forces). It can be described by Cosserat Rod theory and discretized, by modelling it ...
98 views

### Computing discrete laplacian matrix for mesh fairing

I asked this question on the math stack exchange and got an answer, but I am just as utterly confused as before. My fundamental goal is to actually construct the matrix, that is, a series of steps I ...
• 273
38 views

### Constructing generalized Laplacian matrix?

I am staring intently at this paper by Botsch and Kobbelt. In particular, I want to make the matrix specified in equation 5. I am trying to understand the specific computations I must instruct a ...
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1 vote
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### How do people know the classic Lorenz attractor is actually deterministically chaotic at infinite time?

Correct me if I am wrong, but I take it that people actually think the classic Lorenz system is fully characterized, i.e. we know what the attractor looks like and the various fractal dimensions of it,...
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1 vote
153 views

### how to solve a coupled nonlinear partial differential equations(Boundary Value Problem)

So far, I have been looking into linear and nonlinear differential equations(Boundary value problem) that look like the following: $\frac{d^2 \theta}{dz^2} + \sin(\theta) = 0$ I am now familiar with ...
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98 views

### Prediction of sphere (i.e. roast) core temperature heated in an oven

The real-life problem Assume I put a spherical roast with initially constant temperature of start_temp=25 (°C) into an oven with ...
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1 vote
77 views

### Once Lyapunov exponents have converged, can they diverge again?

I know the strength of attraction for a single attractor might vary from place to place. Say, if I calculated the Lyapunov exponents for a small portion of the attractor and they have already ...
• 197
1k views

### Problems solving 2D heat equation using physics-informed neural networks

I am trying to solve 2D heat equation using the physics-informed neural networks approach. The training loss is decreasing, but my final network outputs make no sense. I am using Python/Pytorch. 2D ...
132 views

### finding weak form of nonlinear differential equation for FEM simulation

The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
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### Energy potential involvign pressure and viscoscity?

Making a really long explanation short. I am looking for an energy density function for an incompressible isotropic fluid that involves both viscoscity and pressure. The reason is, there is a way to ...
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### Why is the maximum potential energy greater than the maximum kinetic energy?

I was plotting the energy variation in a mass-spring system. If I define the initial conditions to be at maximum displacement from the origin, the potential energy is plotted correctly but kinetic ...
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1 vote
111 views

### Numerical method for space fractional derivative in 1 dimension

I am very new to the subject of fractional derivatives which arise while characterizing the anomalous transport of passive scalar in turbulence. I have found an equation of the following form, to ...
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