Differential Equation System Solver

Hey ,

I am currently trying to solve a system of differential equations. I create the coefficient matrix as a sparse matrix, but when i try to use the ode solver, it gives me an out of memory error. The code is:

I dont know if it is stiff our not i got to admit, but when just using Tsit5(), or any other non-stiff solver i tried so far mentioned here ODE Solvers DifferentialEquations.jl it ran out of memory as well. So if I use a non-stiff solver and it still runs out of memory then what is the issue there?

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the different solver of the package are sophisticated enough to solve all not-too-exotic problems. I already used a variety of different solvers described in the documentation, but my problem still remains, such that I think that A) is my problem.

Looking at your equations, I can see without running it that the first equations is missing a minus sign. Additionally, since you never set the values for du[3] and du[4], those will keep the default and thus those two components will be constant.

If either of these are not None or non-negative, then theJacobian is assumed to be banded. These give the number oflower and upper non-zero diagonals in this banded matrix.For the banded case, Dfun should return a matrix whoserows contain the non-zero bands (starting with the lowest diagonal).Thus, the return matrix jac from Dfun should have shape(ml + mu + 1, len(y0)) when ml >=0 or mu >=0.The data in jac must be stored such that jac[i - j + mu, j]holds the derivative of the i`th equation with respect to the `j`thstate variable. If `col_deriv is True, the transpose of thisjac must be returned.

The input parameters rtol and atol determine the errorcontrol performed by the solver. The solver will control thevector, e, of estimated local errors in y, according to aninequality of the form max-norm of (e / ewt)

where X (the capital letter variable in the third equation) is the delayed version of x. Then, I'm linking this DDE system to an existing (and larger) ODE system by adding a couple terms to the equations for x and z, and then integrating the combined system all together.

Note however that this creates a jump discontinuity in the first derivative, which when the DDE solver reaches the point of having t - delay near this point, it does not solve it very efficiently unless it knows the exact spot where this discontinuity is (usually you have use an extra option of telling Matlab where it is, but if you follow the steps below, then that will not be needed).

I just wanted to find good Fortran differential solver so that I can use it in my current Fortran code.

I am not going to completely translate my Fortran code to Julia because based on my experience it seems that, it is not very easy to write a Julia code whose speed is the same as Fortran.

Another classic resource are the ODE solvers published at netlib. Unfortunately, most of them are F77 style codes which are not thread-safe. An exception is the rksuite which has a Fortran 90 interface and was later modified and included into the NAG library.

The blog post from Christopher Rackauckas (also listed above by @kargl) lists a few more Fortran codes including ODEPACK, Sundials, and FATODE. Many of these are still heavily used today. Some of the solvers by Hairer and from ODEPACK are wrapped in scipy.integrate.

I remember Sourcery Institute also had a nice differential equation solver library in modern Fortran. I searched their GitHub pages, but could not find it immediately. Perhaps @rouson could help more on this.

An ordinary differential equation (ODE) contains one or more derivatives of a dependent variable, y, with respect to a single independent variable, t, usually referred to as time. The notation used here for representing derivatives of y with respect to t is y' for a first derivative, y'' for a second derivative, and so on. The order of the ODE is equal to the highest-order derivative of y that appears in the equation.

In an initial value problem, the ODE is solved by starting from an initial state. Using the initial condition, y0, as well as a period of time over which the answer is to be obtained, (t0,tf), the solution is obtained iteratively. At each step the solver applies a particular algorithm to the results of previous steps. At the first such step, the initial condition provides the necessary information that allows the integration to proceed. The final result is that the ODE solver returns a vector of time steps t=[t0,t1,t2,...,tf] as well as the corresponding solution at each step y=[y0,y1,y2,...,yf].

If some components of y' are missing, then the equations are called differential algebraic equations, or DAEs, and the system of DAEs contains some algebraic variables. Algebraic variables are dependent variables whose derivatives do not appear in the equations. A system of DAEs can be rewritten as an equivalent system of first-order ODEs by taking derivatives of the equations to eliminate the algebraic variables. The number of derivatives needed to rewrite a DAE as an ODE is called the differential index. The ode15s and ode23t solvers can solve index-1 DAEs.

Fully implicit ODEs of the form f(t,y,y')=0. Fully implicit ODEs cannot be rewritten in an explicit form, and might also contain some algebraic variables. The ode15i solver is designed for fully implicit problems, including index-1 DAEs.

ode45 performs well with most ODE problems and should generally be your first choice of solver. However, ode23, ode78, ode89 and ode113 can be more efficient than ode45 for problems with looser or tighter accuracy requirements.

Some ODE problems exhibit stiffness, or difficulty in evaluation. Stiffness is a term that defies a precise definition, but in general, stiffness occurs when there is a difference in scaling somewhere in the problem. For example, if an ODE has two solution components that vary on drastically different time scales, then the equation might be stiff. You can identify a problem as stiff if nonstiff solvers (such as ode45) are unable to solve the problem or are extremely slow. If you observe that a nonstiff solver is very slow, try using a stiff solver such as ode15s instead. When using a stiff solver, you can improve reliability and efficiency by supplying the Jacobian matrix or its sparsity pattern.

You can use ode objects to automate solver selection based on properties of the problem. If you are not sure which solver to use, then this table provides general guidelines on when to use each solver.

This table contains a list of the available ODE and DAE example files as well as the solvers and options they use. Links are included for the subset of examples that are also published directly in the documentation.

[t,y] =ode45(odefun,tspan,y0),where tspan = [t0 tf], integrates the system ofdifferential equations y'=f(t,y) from t0 to tf withinitial conditions y0. Each row in the solutionarray y corresponds to a value returned in columnvector t.

All MATLAB ODE solvers can solve systems of equations ofthe form y'=f(t,y),or problems that involve a mass matrix, M(t,y)y'=f(t,y).The solvers all use similar syntaxes. The ode23s solveronly can solve problems with a mass matrix if the mass matrix is constant. ode15s and ode23t cansolve problems with a mass matrix that is singular, known as differential-algebraicequations (DAEs). Specify the mass matrix using the Mass optionof odeset.

ode45 is a versatile ODE solver and is thefirst solver you should try for most problems. However, if the problemis stiff or requires high accuracy, then there are other ODE solversthat might be better suited to the problem. See Choose an ODE Solver formore information.

The time step chosen by the solver at each step is based on the equation in the system that needs to take the smallest step. This means the solver can take small steps to satisfy the equation for one initial condition, but the other equations, if solved on their own, would use different step sizes. Despite this, solving for multiple initial conditions at the same time is generally faster than solving the equations separately using a for-loop.

Solve the equation over the time interval [1 5] using ode45. Specify the function using a function handle so that ode45 uses only the first two input arguments of myode. Also, loosen the error thresholds using odeset.

Solve the van der Pol equation with =1 using ode45. The function vdp1.m ships with MATLAB and encodes the equations. Specify a single output to return a structure containing information about the solution, such as the solver and evaluation points.

If tspan has more than two elements [t0,t1,t2,...,tf], then the solver returns the solution evaluated at the given points. However, the solver does not step precisely to each point specified in tspan. Instead, the solver uses its own internal steps to compute the solution, and then evaluates the solution at the requested points in tspan. The solutions produced at the specified points are of the same order of accuracy as the solutions computed at each internal step.

If tspan contains several intermediate points [t0,t1,t2,...,tf], then the specified points give an indication of the scale for the problem, which can affect the value of InitialStep used by the solver. Therefore, the solution obtained by the solver might be different depending on whether you specify tspan as a two-element vector or as a vector with intermediate points.

The initial and final values in tspan are used to calculate the maximum step size MaxStep. Therefore, changing the initial or final values in tspan can cause the solver to use a different step sequence, which might change the solution.

I might write 8 equations by integrating each of the Differential equation 4 times and then substitute the boundary condition and solve the problem by finding unknowns. The program does not automatically catch boundary conditions establishing a relationship of one differential equation to another.

I am attempting to solve and graph the solution to an initial value problem containing a system of differential equations. If I am remembering calculus correctly, its properties (nonlinear, ordinary, no explicit appearance of the independent variable time) classify it as a 'time-invariant autonomous system'.