1. L1 scheme

   Reference:

   • Yanan Zhang, Zhizhong Sun and Honglin Liao. Finite difference methods for the time fractional diffusion equation on non-uniform meshes. Journal of Computational Physics. 265 (2014) 195--210. doi: 10.1016/j.jcp.2014.02.008

2. Fast L1 scheme

   Reference:

   • Shidong Jiang, Jiwei Zhang, Qian Zhang2 and Zhimin Zhang. Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations. Communications in Computational Physics. 21 (2017) 650--678. doi: 10.4208/cicp.OA-2016-0136

3. L1-2 scheme

   Ref:

   • G. Gao et al., J. Comput. Phys., 259(2014) 33--50.

4. L2-1- scheme

   Reference:

   • Anatoly A.Alikhanov. A new difference scheme for the time fractional diffusion equation. Journal of Computational Physics. 280 (2015) 424-438 doi: 10.1016/j.jcp.2014.09.031

   • Hong-lin Liao and William McLean and Jiwei Zhang. A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems. SIAM Journal on Numerical Analysis. 57 (2019) 218-237. doi: 10.1137/16m1175742

5. Fast L2-1- scheme

   Reference:

   • Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme. Communications in Computational Physics. 22 (04) 1028-1048. doi: 10.4208/cicp.OA-2017-0019

These formulas have same input parameters qformula, t_array, u0, tol.

• qformula: This is a structure with two fileds alpha and w, where alpha is(are) the fractional order, and w is(are) the weight of the fractional term. It corresponds to the term of the equation.
• t_array: This is the temporal mesh .
• u0: This is the initial value, it can be any shape: scale or vector or matrix.
• tol: This parameter only used in fast algorithms. It is used to limit the error of the SOE(sum-of-exponentials) approximation.

There are some functions of the formulas.

• update: update the data of the pre step by inputing current step number and pre step data in the for loop. If the current step number is i and the data value of step i-1 is pre_u, we call this function like this [fhp, uh] = fhp.update(i, pre_u); where fhp is the object handle and uh ($u_{hist}$) is the history contribution for the fractional derivative of current step i. See the examples for more details. This function must be call step by step from the beginning.
• get_wn: the weight value of current step data. The product of the return value and the current data if the contribution of current step data for the fractional derivative, i.e. $D^\alpha_{t^{n-\sigma}} u = w u^n u_{hist}$, where $w$ is the return value of get_wn and $u_{hist}$ is the second output of update.
• get_sigma: get the sigma value which can be used to obtain the approximation point . We can obtain it like this $t^{n-\sigma} = \sigma * t^(n-1) (1 - \sigma) * t^n$;
• get_ti: get the mesh value at step i;
• get_tn: get the mesh value at current step.
• get_t: get the approximation point. As the formula approximate the value at , it will return the value of $t^{n - sigma}$.

This is an example to solve the equation , see file example.m.

% solve fractional ordinary differential equation  
%                \[  D^alpha u = f  \]

alpha = 0.8;
c = 3;
u_fun = @(t) t.^c;
f_fun = @(t) gamma(c 1)/gamma(c 1-alpha) * t.^(c-alpha); % source term

m = 100;
T = 1;
qformula.alpha = alpha;
qformula.w = 1;
t_array = T*(0:m)/m;
u0 = u_fun(0);

tol = 1e-8;

formula1 = @L1_formula;
formula2 = @L1_2_formula_uniform;
formula3 = @L2_1_sigma_single_term;

formula4 = @Fast_L1_formula;
formula5 = @Fast_L2_1_sigma_uniform;
formula6 = @Fast_L2_1_sigma_single_term;

formula = formula6;
fhp = formula(qformula, t_array, u0, tol);

un_pre = u0; % 
u = zeros(m, 1);
for i = 1:m
    % update fhp by add the previous value of u.  un_pre = u(i-1);
    [fhp, uh] = fhp.update(i, un_pre); % According to matlab's syntex, we must return the value of the object here.
                 
    % get the approximation point as follows, or you can get the
    % approximation point just call  t = fhp.get_t();
    sigma = fhp.get_sigma();
    t_i_minus_1 = fhp.get_ti(i-1);  % get t^(i-1)
    t_i = fhp.get_ti(i);            % get t^i
    t = sigma * t_i_minus_1   (1-sigma) * t_i; % t^{i-sigma} = sigma * t^(i-1)   (1 - sigma) * t^i;
    
    b = f_fun(t);
    c0 = fhp.get_wn(i); % get the coefficient of u(i);
    % we have fhp.get_wn(i) * u(i)   uh = f
    u(i) = (b - uh)/c0;
    un_pre = u(i);
end
u_exact = u_fun(t_array(2:end));
plot(t_array(2:end), u, '-', t_array(2:end), u_exact, 'x');
legend({'Numerical Result', 'Exact Solution'});