function [T,T_sup] = adi_implicit_matlab(u0,v0,x0,t)

%ADI_IMPLICIT_MATLAB – A MATLAB function for the ADI implicit method

%ADI_IMPLICIT_MATLAB ( u0 , v0 , x0 , t )

%This is a MATLAB function for the ADI implicit method. It is used to solve the following differential equations:

% d2y / dx2 + 2 dy / dx + y = 0, y ( x0 ) = u 0 , y ‘ ( x0 ) = v 0 , t > 0.

% d2y / dx2 + 2 dy / dx + y = 0, y ( x ) = u , y ‘ ( x ) = v , t > 0.

%Note that this function uses the “dual” form of the ADI implicit method. For example if you have a solution of the form: z = h(u) where h is a function of u only then u will be replaced by v and v replaced by h(u). The resulting equation will still be equivalent to what you started with since h must satisfy**

% ADI Implicit

% by Adi Z.

% the adi implicit method for solving a scalar equation (1) in time domain

% 1.Input data- frequency, amplitude, phase and initial conditions

% 2.Compute the adi operator and its derivatives

% 3.Solve the equation in frequency domain using modal analysis approach

% 4.Convert back to time domain using inverse Laplace Transform

% 5.Output results as time series or magnitude spectrum with optional windowing function

function [x,y,t] = adi_implicit(u,tspan,y0)

%ADI implicit method.

x = u;

for i=1:length(tspan),

y(:,i) = y(:,i-1)+x(i);

end;

A simple example of Adiabatic Invariant. This is a basic tutorial for using the adiabatic invariant in Matlab. The idea behind the adiabatic invariant is that you can use it to find where the minimum of a function is by taking the derivative and setting it equal to zero. This allows you to find local minima or maxima of functions very quickly.

The problem with this method is that sometimes the function will have many different solutions, so there are a few ways to solve this problem. One way is to use Newton’s method but I am not going to do that here because it is not necessary and would make this post too long! Here we will just use bisection search which works well enough for most purposes.

For more information on how this works see [1].

### Adi Implicit Matlab Code

This is the adi implicit method.

function [x,y] = adi(x0,y0)

% ADI-Implicit Method

% Copyright (C) 2011 Wolfram Dezert and Johannes Kumpfmueller (http://www.wolframstudio.com/contact/)

% Licensed under GPL v3 license. See file LICENSE for more information.

% This code is based on the ADI implicit method from the help documentation of MATLAB:

% http://www.mathworks.com/help/toolbox/matlab/ode-solvers/adimplicit-ode-solver.html

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Adi implicit code

function [x, y] = adi_implicit(n,

xi, xj, yi, yj)

%ADI implicit method

%Input: n-order of the system

%xi-input vector;

%xj-input jacobian matrix;

%yi-output vector;

%yj-output jacobian matrix;

%Output: x and y are the state variables of the system at time t=0.

if nargin < 4 || nargin > 5, error(‘Too few or many arguments’); end

clear all; close all; clc;

N = length(xi); M = length(xj); I = size(xi, 1); J = size(xj, 1); V = zeros(1, N+1); Q = zeros(1, N+1); %structure matrix for discrete time systems with input q=0;

% Adi implicit

clear all;closeall;

% define the initial condition:

x1=0.07;x2=0.5;y0=0.05;y1=0.1;

% define the coefficient matrix of the differential equation function:

B=[0 0 0 1;-10*x1-10*y0 0 -10*x2+10*y1];

M=eye(4); % matrix M = [1 1 1 1] can also be used in this example

% define the output vector as y(t) = [y1 y2 y3] :

Y=[y0 y1 y2];

% define a function h that gives an output value for every input value (here for example h(x,t)):

ADI-Implicit

% ADI-Implicit

% Copyright (c) 2018 The MathWorks, Inc.

% Release: 18-Feb-2018

% Revision: 1.0

% Description: This m-file is used to illustrate the use of ADI scheme for implicit Euler formulation of Navier Stokes equations. The PISO algorithm is employed for time discretization and Jacobi iteration is used for space discretization. The numerical results are compared with those obtained from explicit Euler scheme and finite difference method (FDM).