How can I solve the following four coupled equations numerically in Python?
I need a source code !!!!!
y'(1) = (-wa+wp)*y(2)+g*y(4)
y'(2) = (-wa+2wp)*y(1)-g*y(3)
y'(3) = (wa-wp)*y(4)+g*y(2)
y'(4) =-(wa+wp)*y(3)-g*y(1)
Check out Integration and ODEs (scipy.integrate). Exactly which solver you use will depend upon the boundary conditions you have: particularly solve_ivp for initial value problems and solve_bvp for boundary value problems.
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I am working on ordinary differential equations that I want to solve using numerical methods in Python. Some of these ODEs are chemical equations, e.g. A + B > C with stoichiometry coefficient sigma. This leads to the differential equation du/dt = sigma * a * b.
To do this, the classical method is to use e.g. a Runge-Kutta solver, such as the one in Scipy's integrate.ode.
Instead, I would like to use a modeling language similar to cvxpy is for convex optimization. This method would allow to formally define the ODE and automatically solve the equation. It would be similar, in Python, to what is done in OpenModelica.
Is there such a tool?
If I have a system of nonlinear ordinary differential equations, M(t,y) y' = F(t,y), what is the best method of solution when my mass matrix M is sometimes singular?
I'm working with the following system of equations:
If t=0, this reduces to a differential algebraic equation. However, even if we restrict t>0, this becomes a differential algebraic equation whenever y4=0, which I cannot set a domain restriction to avoid (and is an integral part of the system I am trying to model). My only previous exposure to DAEs is when an entire row is 0 -- but in this case my mass matrix is not always singular.
What is the best way to implement this numerically?
So far, I've tried using Python where I add a small number (0.0001) to the main diagonals of M and invert it, solving the equations y' = M^{-1}(t,y) F(t,y). However, this seems prone to instabilities, and I'm unsure if this is a universally appropriate means of regularization.
Python doesn't have any built-in functions to deal with mass matrices, so I've also tried coding this in Julia. However, DifferentialEquations.jl states explicitly that "Non-constant mass matrices are not directly supported: users are advised to transform their problem through substitution to a DAE with constant mass matrices."
I'm at a loss on how to accomplish this. Any insights on how to do this substitution or a better way to solve this type of problem would be greatly appreciated.
The following transformation leads to a constant mass matrix:
.
You need to handle the case of y_4 = 0 separately.
How can I solve a system of k differential equations with derivatives appearing in every equation? I am trying to use Scipy's solve_ivp.
All the equations are of the following form:
equations
How can this system of equations be numerically solved using any solver? using solve_ivp, it seems you should be able to write every equation independent of the other ones, which seems not possible in this case when we have more than 2 equations.
If you set C[i]=B[i,i] then you can transform the equations to the linear system B*z'=A. This can be solved as
zdot = numpy.linalg.solve(B,A)
so that the derivative is this constant solution of a constant linear system, and the resulting solution for z is linear, z(t)=z(0)+zdot*t.
I'm on the lookout for a numerical method that can solve both a deterministic and stochastic equation. In the deterministic case, I know that a fourth order RK method is a valuable one, very effective. Unfortunately, there has not been applied to stochastic equations successfully (at least as far as I know).
Now what I want to know is if a numerical method that can solve both equations (roughly I mean, in comparison to the analytic solutions) exists and, in that case, what would be. A stochastic equation analytically solvable would be the Black-Scholes one, for instance.
There are methods for solving these kinds of equations in DifferentialEquations.jl. Stochastic differential equations are a form of mixed deterministic and stochastic equation and solving them is shown in the SDE tutorial. Mixing discrete stochasticity with deterministic equations is shown in the jump equation tutorial. While written naively in Julia, it is accessible in Python via the package diffeqpy. Notice that this has some example stochastic differential equations in the README.
I have a following
ordinary differential equation
and numeric parameters Sigma=0.4, x(0) = 4 and dx(0)/dt = 0
My task is to get Cauchy problem solution (Initial value problem solution) of differential equation using ode function
Can someone help me? I don't even know how to write equation and especially numeric parameters in correct way for SciPy.
P.S. Sorry for not posting images, I've just registered.
Like Warren said, scipy.integrate.odeint is the 'SciPy' way to solve this.
But before you take your problem to SciPy (or whatever solver you end up using) you'll want to convert your 2nd order ODE to a first order ODE using something like: http://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx
To get things into SciPy you need to get your equation looking like:
y' = f(y)
But right now your equation is written like:
y'' = f(y, y')
The solution is to add more variables to your system, but the link will explain it more thoroughly.