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5 Differential Equations Troubleshooting (Continued) Question I’m plotting the solution as it is computed and it looks fine, but the code gets stuck at some point. Answer First verify that the ODE function is smooth near the point where the code gets stuck. If it isn’t, the solver must take small steps to deal with this. It may help to break tspan into pieces on which the ODE function is smooth. If the function is smooth and the code is taking extremely small steps, you are probably trying to solve a stiff problem with a solver not intended for this purpose. Switch to ode15s, ode23s, ode23t, or ode23tb. My integration proceeds very slowly, using too many time steps. First, check that your tspan is not too long. Remember that the solver uses as many time points as necessary to produce a smooth solution. If the ODE function changes on a time scale that is very short compared to the tspan, the solver uses a lot of time steps. Long-time integration is a hard problem. Break tspan into smaller pieces. If the ODE function does not change noticeably on the tspan interval, it could be that your problem is stiff. Try using ode15s, ode23s, ode23t, or ode23tb. Finally, make sure that the ODE function is written in an efficient way. The solvers evaluate the derivatives in the ODE function many times. The cost of numerical integration depends critically on the expense of evaluating the ODE function. Rather than recompute complicated constant parameters at each evaluation, store them in globals or calculate them once and pass them to nested functions. I know that the solution undergoes a radical change at time t where t0 ≤ t ≤ tf but the integrator steps past without “seeing” it. If you know there is a sharp change at time t, it might help to break the tspan interval into two pieces, [t0 t] and [t tf], and call the integrator twice. If the differential equation has periodic coefficients or solution, you might restrict the maximum step size to the length of the period so the integrator won’t step over periods. 5-48
Initial Value Problems for DDEs Initial Value Problems for DDEs This section describes how to use MATLAB to solve initial value problems (IVPs) for delay differential equations (DDEs). It provides: • A summary of the DDE functions and examples • An introduction to DDEs • A description of the DDE solver and its syntax • General instructions for representing a DDE • A discussion and example about discontinuities and restarting • A discussion about changing default integration properties DDE Function Summary DDE Initial Value Problem Solver Solver dde23 Description Solve initial value problems for delay differential equations with constant delays. DDE Helper Functions Function deval Description Evaluate the numerical solution using the output of dde23. DDE Solver Properties Handling An options structure contains named properties whose values are passed to dde23, and which affect problem solution. Use these functions to create, alter, or access an options structure. 5-49
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MATLAB® The Language of Technical
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2 Polynomials and Interpolation Pol
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Introduction to Initial Value ODE P
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1 Matrices and Linear Algebra Funct
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3 Fast Fourier Transform (FFT) Intr
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3 Fast Fourier Transform (FFT) 200
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3 Fast Fourier Transform (FFT) 2 x
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3 Fast Fourier Transform (FFT) 500
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4 Function Functions Function Summa
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6 Sparse Matrices Function Summary
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6 Sparse Matrices Function Summary
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6 Sparse Matrices This matrix requi
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6 Sparse Matrices S = (3,1) 1 (2,2)
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6 Sparse Matrices Now F = full(S) d
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6 Sparse Matrices Importing Sparse
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6 Sparse Matrices west0479 west0479
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6 Sparse Matrices The find Function
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6 Sparse Matrices of the rows and c
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6 Sparse Matrices 0 500 1000 1500 2
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6 Sparse Matrices simply sparse(m,n
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6 Sparse Matrices Similarly, S(:,p)
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6 Sparse Matrices Manipulating Spar
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6 Sparse Matrices Selected Bibliogr
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Index comparing sparse and full mat
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Index H hb1dae demo 5-35 hb1ode dem
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Index nonstiff ODE examples rigid b
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Index LU factorization 6-30 minimum
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Index twobvp demo 5-63 two-dimensio
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