Questions tagged [ode]

Ordinary Differential Equations (ODEs) contain functions of only one independent variable, and one or more of their derivatives with respect to that variable. This tag is intended for questions on modeling phenomena with ODEs, solving ODEs, and other related aspects.

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A non linear ode with boundary conditions at infinity

I want to solve the non-linear ODE $$\frac{d^2}{dx^2}y=a(y+y^3)$$ With the boundary conditions that $$\lim_{x\to \pm \infty} y(x) =0$$ I am not aware of any analytical method for solving this kind ...
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Can Taylor methods be used effectively on stiff ODEs?

Cleve Moler has stated that "all numerical methods for stiff odes are implicit." However, I don't know whether this statement is a mathematical fact, or an simply an observation. Moreover, many ...
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Solving systems of nonlinear ODEs in epidemiology

I have a 9 systems of nonlinear ODEs to solve. I want to determine the endemic equilibrium points. How do I go about it? I tried manual calculation but it becomes cumbersome. Can software be used ...
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Dormand–Prince 5(4): How to update the stepsize and make accept/reject decision?

https://en.wikipedia.org/wiki/Dormand–Prince_method I want to implement the Dormand-Prince 4(5) version to solve Initial Value problems. Using regular notation I have $A$ matrix and the $c,b,\hat{b}$ ...
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Is there a database/website with Butcher tableaus?

I have started investigating in mostly Runge Kutta and Runge Kutta Nyström methods and there one of the only differences between the methods of the same type is their Butcher tableu. For the most ...
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The eigenvalue problem $$\frac{d^2u}{dx^2}+2i k\frac{du}{dx}-[k^2-6\sin(x)^2]u(x)=-\mu u(x)$$ gives the first five eigenvalues with $k=0$ or $k=1$ which are $2.06$, $2.26$, $5.16$, $6.81$, and 7.... 0answers 61 views How to implement adaptive step size Runge-Kutta Cash-Karp? Trying to implement an adaptive step size Runge-Kutta Cash-Karp but failing with this error: ... 1answer 97 views Formulation of the least-squares parameter estimation problem I have a system of 10 ordinary differential equations of the form, $$\frac{dy_1}{dt} = f1(V1,k1,y1,y2)\\ \vdots \\ \frac{dy_{10}}{dt} = f_{10}(V_{10},k_{10},y_{9},y_{10})$$ I want to estimate the ... 0answers 39 views Time sampling changes solution I'm currently trying to solve a problem using numerical methods. The set-up is rather long, so I apologize in advance... TL;DR: My solutions change depending on how big my steps are and I don't know ... 0answers 64 views Solve ODE with non-negative and maximization constraints My task is to solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$ $$C_k\ge0$$ $$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ with e_1 = -\beta_1-\beta_3... 1answer 89 views What's the minimum step size that can be used in Euler's method before it becomes unreliable? In particular, if Euler's method is implemented on a computer, what's the minimum step size that can be used before rounding errors cause the Euler approximations to become completely unreliable? I ... 1answer 141 views Solving differential equation in Python with variable coefficients (I just know the coefficients numerically) I am trying to implement a routine to solve a differential equation in Python. Basically the kind of equation that I am interested in solving is of the form: \displaystyle \frac{d}{dx^2} \left(x y(x)... 1answer 413 views What is the state of the art in solving stiff initial value problems? I'm looking for current references on solving stiff ODEs. Most of what I know (say, BDF methods) apparently date back to the 1980's, and I feel like a lot of progress should have been made in that ... 1answer 159 views Mass Matrix and how to handle it (ODEs) - References I'm interested in a good reference/paper about how to handle numerically a mass-matrix system as \begin{align} \mathbf{M}(t,y)\dot{y} =F(y,t) \end{align} I know that such a problem can be solved by ... 2answers 134 views Non-linear Boundary Value Problem. How to compute the Jacobian? Consider a Boundary Value Problem: \delta u''+u(u'-1) =0 \Leftrightarrow u''=\frac{-u(u'-1)}{\delta}=:f(t,u',u), \\ u(0)=a, u(1)=b\delta,a,b$are known parameters. I want to implement Newton'... 1answer 80 views Combining multiple coupled 1st order equations in python I'm having serious troubles with solving translating 3 coupled differential equations into python. The 3 DE's stem from a 4th order DE used to calculate the bending moment of an underwater pipeline ... 1answer 128 views What does the exponential function mean in numerical ODE solving formulas? I'm trying to read papers on numerical ODE algorithms and I always seem to stumble upon huge amounts of exponentials multiplied by each other. For example in New families of symplectic splitting ... 0answers 17 views Translating grid with extrusion speed I am putting into MATLAB code the equations that describe a plastic extrusion process. From a paper, I found I should use a spatial grid that translates with the extrusion speed, being the reference ... 1answer 137 views Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous? Existing algorithms for solving ODEs handle functions$\frac{dy}{dt} = f(y, t)$, where$y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so$\frac{dy}{dt} = f(...
When solving a system of ODEs, as part of a preconditioner, I get the system $(A + D(t))x = b(t)$ where $A$ is a sparse matrix and $D(t)$ is diagonal. I'm currently solving this by taking the LU-...