MATLAB Mathematics - SERC - Index of

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5 Differential Equations 3 Code the boundary conditions function. You must also code the boundary conditions in a function. Because there is an unknown parameter, the function must be of the form res = bcfun(ya,yb,parameters) The code below represents the boundary conditions in the function, mat4bc. function res = mat4bc(ya,yb,lambda) res = [ ya(2) yb(2) ya(1)-1 ]; 4 Create an initial guess. To form the guess structure solinit with bvpinit, you need to provide initial guesses for both the solution and the unknown parameter. The function mat4init provides an initial guess for the solution. mat4init uses y = cos4x because this function satisfies the boundary conditions and has the correct qualitative behavior (the correct number of sign changes). function yinit = mat4init(x) yinit = [ cos(4*x) -4*sin(4*x) ]; In the call to bvpinit, the third argument, lambda, provides an initial guess for the unknown parameter λ . lambda = 15; solinit = bvpinit(linspace(0,pi,10),@mat4init,lambda); This example uses @ to pass mat4init as a function handle to bvpinit. Note See the function_handle (@), func2str, and str2func reference pages, and the “Function Handles” chapter of “Programming and Data Types” in the MATLAB documentation for information about function handles. 5 Apply the BVP solver. The mat4bvp example calls bvp4c with the functions mat4ode and mat4bc and the structure solinit created with bvpinit. sol = bvp4c(@mat4ode,@mat4bc,solinit); 5-70
Boundary Value Problems for ODEs 6 View the results. Complete the example by displaying the results: a Print the value of the unknown parameter λ found by bvp4c. fprintf('The fourth eigenvalue is approximately %7.3f.\n',... sol.parameters) b Use deval to evaluate the numerical solution at 100 equally spaced points in the interval [ 0, π] , and plot its first component. This component approximates yx ( ). xint = linspace(0,pi); Sxint = deval(sol,xint); plot(xint,Sxint(1,:)) axis([0 pi -1 1.1]) title('Eigenfunction of Mathieu''s equation.') xlabel('x') ylabel('solution y') See “Evaluating the Solution at Specific Points” on page 5-72 for information about using deval. The following plot shows the eigenfunction associated with the final eigenvalue λ = 17.097. solution y 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 Eigenfunction of Mathieu’s equation. −1 0 0.5 1 1.5 2 2.5 3 x 5-71
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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 Singu
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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 Cal
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2 Polynomials and Interpolation Int
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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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5 Differential Equations Initial Va
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5 Differential Equations odephas3 o
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5 Differential Equations This secti
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5 Differential Equations The basic
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5 Differential Equations 2 Code the
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5 Differential Equations function.
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5 Differential Equations One way to
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5 Differential Equations y0, yp0 Ve
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5 Differential Equations For exampl
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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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