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5 Differential Equations The example solves the equations on [0,5] with history y 1 () t = 1 y 2 () t = 1 y 3 () t = 1 for t ≤ 0 . Note The demo ddex1 contains the complete code for this example. To see the code in an editor, click the example name, or type edit ddex1 at the command line. To run the example type ddex1 at the command line. See “DDE Solver Basic Syntax” on page 5-52 for more information. 1 Rewrite the problem as a first-order system. To use dde23, you must rewrite the equations as an equivalent system of first-order differential equations. Do this just as you would when solving IVPs and BVPs for ODEs (see “Examples: Solving Explicit ODE Problems” on page 5-9). However, this example needs no such preparation because it already has the form of a first-order system of equations. 2 Identify the lags. The delays (lags) τ 1 , …, τ k are supplied to dde23 as a vector. For the example we could use lags = [1,0.2]; In coding the differential equations, τ j = lags(j). 3 Code the system of first-order DDEs. Once you represent the equations as a first-order system, and specify lags, you can code the equations as a function that dde23 can use. This code represents the system in the function, ddex1de. function dydt = ddex1de(t,y,Z) ylag1 = Z(:,1); ylag2 = Z(:,2); 5-54
Initial Value Problems for DDEs dydt = [ylag1(1) ylag1(1) + ylag2(2) y(2) ]; 4 Code the history function. The history function for this example is function S = ddex1hist(t) S = ones(3,1); 5 Apply the DDE solver. The example now calls dde23 with the functions ddex1de and ddex1hist. sol = dde23(@ddex1de,lags,@ddex1hist,[0,5]); Here the example supplies the interval of integration [0,5] directly. Because the history is constant, we could also call dde23 by sol = dde23(@ddex1de,lags,ones(3,1),[0,5]); 6 View the results. Complete the example by displaying the results. dde23 returns the mesh it selects and the solution there as fields in the solution structure sol. Often, these provide a smooth graph. plot(sol.x,sol.y); title('An example of Wille'' and Baker'); xlabel('time t'); ylabel('solution y'); legend('y_1','y_2','y_3',2) 5-55
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MATLAB® The Language of Technical
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Revision History June 2004 First pr
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2 Polynomials and Interpolation Pol
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Introduction to Initial Value ODE P
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vi Contents
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1 Matrices and Linear Algebra Funct
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1 Matrices and Linear Algebra Matri
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1 Matrices and Linear Algebra Addin
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1 Matrices and Linear Algebra If x
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1 Matrices and Linear Algebra Eigen
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1 Matrices and Linear Algebra 1-44
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2 Polynomials and Interpolation Pol
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2 Polynomials and Interpolation whe
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2 Polynomials and Interpolation 2-3
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2 Polynomials and Interpolation 2-3
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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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3 Fast Fourier Transform (FFT) Func
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4 Function Functions Function Summa
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4 Function Functions You can also c
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4 Function Functions 25 20 15 10 5
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4 Function Functions Minimizing Fun
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4 Function Functions To try fminsea
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4 Function Functions Plotting the R
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4 Function Functions The number of
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4 Function Functions and displays t
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4 Function Functions end function s
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4 Function Functions 100 80 60 40 2
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4 Function Functions You can verify
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4 Function Functions Troubleshootin
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4 Function Functions A three-dimens
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4 Function Functions Parameterizing
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4 Function Functions “Anonymous F
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5 Differential Equations Initial Va
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5 Differential Equations u1(x,t) 1
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5 Differential Equations Selected B
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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 The vertices of o
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6 Sparse Matrices 0 10 20 30 40 50
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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 The following MAT
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6 Sparse Matrices 0 Original 0 Reve
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6 Sparse Matrices QR Factorization
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6 Sparse Matrices shows that A has
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6 Sparse Matrices Functions for Ite
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6 Sparse Matrices set up the five-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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