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DIFFERENTIAL EQUATIONS, INTEGRATION, POLYNOMIALS, RANDOM NUMBERS

DIFFERENTIAL EQUATIONS, INTEGRATION, POLYNOMIALS, RANDOM NUMBERS. MEKANIKA SOFTWARE SEMINARS. ODE INITIAL VALUE PROBLEMS. ODE solvers handle the following types of first-order ODE’s: Explicit ODEs Linearly implicit ODEs Fully implicit ODEs ODE IVP solvers include:

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DIFFERENTIAL EQUATIONS, INTEGRATION, POLYNOMIALS, RANDOM NUMBERS

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  1. DIFFERENTIAL EQUATIONS, INTEGRATION, POLYNOMIALS, RANDOM NUMBERS MEKANIKA SOFTWARE SEMINARS

  2. ODE INITIAL VALUE PROBLEMS • ODE solvers handle the following types of first-order ODE’s: • Explicit ODEs • Linearly implicit ODEs • Fully implicit ODEs • ODEIVP solvers include: • ode45, ode23, ode113 (non-stiff DE’s) • ode23t, ode23tb (moderately stiff DE’s) • ode15s, ode23s (stiff DE’s) • ode15i (fully implicit DE’s) • All ODE’s must be reduced to first order DE’s by substitutions before MATLAB can actually solve them.

  3. ODE IVP SYNTAX • ODE syntax : • [t,y] = solver (odefun , tspan , y0) • [t,y] = ode15i (odefun , tspan, y0, yp0) • The odefun is generally given as a function handle: e.g. @function, where function defines the ODE. • Another parameter may be included in certain cases in the syntax, viz. options • For ode15i one must endeavor to use the decic function in MATLAB to compute consistent initial conditions. [y0mod,yp0mod] = decic (odefun,t0,y0,fixed_y0,yp0,fixed_yp0)

  4. ODE BOUNDARY VALUE PROBLEMS • Just as IVP’s require the initial conditions to be explicitly specified, BVP’s require boundary conditions to be specified. • 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: solinit = bvpinit( x, yinit, parameters) • This generates an initial mesh from which bvpinit can form a good starting point for the BVP solver • The BVP solver is bvp4c, which solves 2-point BVP’s using a 3-stage finite difference Lobatto-Illa formula which is 4th order uniformly accurate.

  5. ODE BVP SYNTAX • sol = bvp4c(odefun,bcfun,solinit) • bcfun is a function that computes the residual in the boundary conditions. • res = bcfun (ya,yb) • solinit is a structure containing the initial guess for a solution.

  6. PARTIAL DIFFERENTIAL EQUATIONS • c * δu/δt = x-m * f + s • c : coupling matrix; diagonal • m : symmetry of the problem • f : flux vector • s : source vector • all of the above are expressed as some function of (x, t, u, δu/δx) • The PDE holds for t0<t<tfand a<x<b and must satisfy : • IC :u(x, t0) = u0(x) • BC : p(x,t,u) + q(x,t)*f = 0

  7. PDE SOLVER SYNTAX • sol = pdepe (m,pdefun,icfun,bcfun,xmesh,tspan) • [c,f,s] = pdefun (x,t,u,dudx) • u = icfun(x) • [pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t) • Vector [x0, x1, ..., xn] • Vector [t0, t1, ..., tf] • 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) • [uout ,DuoutDx] = pdeval (m,x,u(j,:),xout)

  8. NUMERICAL INTEGRATION • Numerically evaluate with adaptive Simpson quadrature • q = quad(fun,a,b,tol) • Numerically evaluate with adaptive Lobatto quadrature • q = quadl(fun,a,b,tol) • Double integration • q = dblquad(fun,xmin,xmax,ymin,ymax,tol) • Triple Integration • triplequad(fun,xmin,xmax,ymin,ymax,zmin,zmax)

  9. IMPORTANT TRANSFORMS • FOURIER TRANSFORM : F = fourier(f) • LAPLACE TRANSFORM : L = laplace(F) • Z-TRANSFORM : F = ztrans(f) • INVERSE FOURIER : F = ifourier(f) • INVERSE LAPLACE : L = ilaplace(F) • INVERSE Z-TRANSFORM : F = iztrans(f)

  10. POLYNOMIAL MANIPULATION • CONVOLUTION : w = conv(u,v) • DECONVOLUTION : [q,r] = deconv(v,u) • RESIDUE : [r,p,k] = residue(b,a) • DERIVATIVE : k = polyder(p) • INTEGRATION : polyint(p,k) • CURVE FITTING : [p,S,mu] = polyfit(x,y,n) • EVALUATION : [y,delta] = polyval(p,x,S,mu)

  11. RANDOM NUMBER GENERATION • UNIFORM RANDOM NUMBERS : Y = rand(m,n) • NORMAL RANDOM NUMBERS : Y = randn(m,n) • RANDOM PERMUTATION : p = randperm(n)

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