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Matlab II

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Mekaniaka Software seminar on Matlab.
Differential Equations, Integration, Polynomials, Random Numbers etc

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Matlab II

  1. 1. DIFFERENTIAL EQUATIONS, INTEGRATION, POLYNOMIALS, RANDOM NUMBERS MEKANIKA SOFTWARE SEMINARS
  2. 2. ODE INITIAL VALUE PROBLEMS <ul><li>ODE solvers handle the following types of first-order ODE’s: </li></ul><ul><ul><li>Explicit ODEs </li></ul></ul><ul><ul><li>Linearly implicit ODEs </li></ul></ul><ul><ul><li>Fully implicit ODEs </li></ul></ul><ul><li>ODE IVP solvers include: </li></ul><ul><ul><li>ode45 , ode23, ode113 (non-stiff DE’s) </li></ul></ul><ul><ul><li>ode23t, ode23tb (moderately stiff DE’s) </li></ul></ul><ul><ul><li>ode15s , ode23s (stiff DE’s) </li></ul></ul><ul><ul><li>ode15i (fully implicit DE’s) </li></ul></ul><ul><li>All ODE’s must be reduced to first order DE’s by substitutions before MATLAB can actually solve them. </li></ul>
  3. 3. ODE IVP SYNTAX <ul><li>ODE syntax : </li></ul><ul><ul><li>[t,y] = solver (odefun , tspan , y0) </li></ul></ul><ul><ul><li>[t,y] = ode15i (odefun , tspan, y0, yp0) </li></ul></ul><ul><li>The odefun is generally given as a function handle: e.g. @function, where function defines the ODE. </li></ul><ul><li>Another parameter may be included in certain cases in the syntax, viz. options </li></ul><ul><li>For ode15i one must endeavor to use the decic function in MATLAB to compute consistent initial conditions. </li></ul><ul><li>[y0mod,yp0mod] = decic (odefun,t0,y0,fixed_y0,yp0,fixed_yp0) </li></ul>
  4. 4. ODE BOUNDARY VALUE PROBLEMS <ul><li>Just as IVP’s require the initial conditions to be explicitly specified, BVP’s require boundary conditions to be specified. </li></ul><ul><li>While solving BVP’s we must give MATLAB a consistent and continuous initial solution from where to begin its iterations. To do this, we use the bvpinit function as: </li></ul><ul><li>solinit = bvpinit( x, yinit, parameters) </li></ul><ul><li>This generates an initial mesh from which bvpinit can form a good starting point for the BVP solver </li></ul><ul><li>The BVP solver is bvp4c, which solves 2-point BVP’s using a 3-stage finite difference Lobatto-Illa formula which is 4 th order uniformly accurate. </li></ul>
  5. 5. ODE BVP SYNTAX <ul><li>sol = bvp4c(odefun,bcfun,solinit) </li></ul><ul><li>bcfun is a function that computes the residual in the boundary conditions. </li></ul><ul><ul><li>res = bcfun (ya,yb) </li></ul></ul><ul><li>solinit is a structure containing the initial guess for a solution. </li></ul>
  6. 6. PARTIAL DIFFERENTIAL EQUATIONS <ul><li>c * δ u/ δ t = x -m * f + s </li></ul><ul><ul><li>c : coupling matrix; diagonal </li></ul></ul><ul><ul><li>m : symmetry of the problem </li></ul></ul><ul><ul><li>f : flux vector </li></ul></ul><ul><ul><li>s : source vector </li></ul></ul><ul><ul><li>all of the above are expressed as some function of </li></ul></ul><ul><ul><li>(x, t, u, δ u/ δ x) </li></ul></ul><ul><li>The PDE holds for t 0 <t<t f and a<x<b and must satisfy : </li></ul><ul><ul><li>IC : u(x, t 0 ) = u 0 (x) </li></ul></ul><ul><ul><li>BC : p(x,t,u) + q(x,t)*f = 0 </li></ul></ul>
  7. 7. PDE SOLVER SYNTAX <ul><li>sol = pdepe (m,pdefun,icfun,bcfun,xmesh,tspan) </li></ul><ul><ul><li>[c,f,s] = pdefun (x,t,u,dudx) </li></ul></ul><ul><ul><li>u = icfun(x) </li></ul></ul><ul><ul><li>[pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t) </li></ul></ul><ul><ul><li>Vector [x0, x1, ..., xn] </li></ul></ul><ul><ul><li>Vector [t0, t1, ..., tf] </li></ul></ul><ul><li>sol is a 3-dimensional vector where sol(i , j , k) gives component k of the solution at time tspan(i) and the mesh point xmesh(j) </li></ul><ul><li>[uout ,DuoutDx] = pdeval (m,x,u(j,:),xout) </li></ul>
  8. 8. NUMERICAL INTEGRATION <ul><li>Numerically evaluate with adaptive Simpson quadrature </li></ul><ul><ul><li>q = quad(fun,a,b,tol) </li></ul></ul><ul><li>Numerically evaluate with adaptive Lobatto quadrature </li></ul><ul><ul><li>q = quadl(fun,a,b,tol) </li></ul></ul><ul><li>Double integration </li></ul><ul><ul><li>q = dblquad(fun,xmin,xmax,ymin,ymax,tol) </li></ul></ul><ul><li>Triple Integration </li></ul><ul><ul><li>triplequad(fun,xmin,xmax,ymin,ymax,zmin,zmax) </li></ul></ul>
  9. 9. IMPORTANT TRANSFORMS <ul><li>FOURIER TRANSFORM : F = fourier(f) </li></ul><ul><li>LAPLACE TRANSFORM : L = laplace(F) </li></ul><ul><li>Z-TRANSFORM : F = ztrans(f) </li></ul><ul><li>INVERSE FOURIER : F = ifourier(f) </li></ul><ul><li>INVERSE LAPLACE : L = ilaplace(F) </li></ul><ul><li>INVERSE Z-TRANSFORM : F = iztrans(f) </li></ul>
  10. 10. POLYNOMIAL MANIPULATION <ul><li>CONVOLUTION : w = conv(u,v) </li></ul><ul><li>DECONVOLUTION : [q,r] = deconv(v,u) </li></ul><ul><li>RESIDUE : [r,p,k] = residue(b,a) </li></ul><ul><li>DERIVATIVE : k = polyder(p) </li></ul><ul><li>INTEGRATION : polyint(p,k) </li></ul><ul><li>CURVE FITTING : [p,S,mu] = polyfit(x,y,n) </li></ul><ul><li>EVALUATION : [y,delta] = polyval(p,x,S,mu) </li></ul>
  11. 11. RANDOM NUMBER GENERATION <ul><li>UNIFORM RANDOM NUMBERS : Y = rand(m,n) </li></ul><ul><li>NORMAL RANDOM NUMBERS : Y = randn(m,n) </li></ul><ul><li>RANDOM PERMUTATION : p = randperm(n) </li></ul>

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