init

FOTF Toolbox/nlfode_mat.m

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+function [y,t]=nlfode_mat(f,alpha,y0,tn,h,p,yp)
+% nlfode_mat - O(h^p) predictor solution of nonlinear single-term FODEs
+%
+%    [y,t]=nlfode_mat(f,alpha,y0,tn,h,p,yp)
+%
+%   f - function handles for the nonlinear explicit FODE
+%   alpha - the maximum order
+%   y0 - initial vector of the output and its integer-order derivatives
+%   tn - terminate time in the solution
+%   h - fixed step-size
+%   p - the order for precision
+%   yp - the predictor solution
+%   y, t - the solution vector and time vector
+
+% Copyright (c) Dingyu Xue, Northeastern University, China
+% Last modified 28 March, 2017 
+% Last modified 18 May, 2022 
+   arguments
+      f, alpha(1,:) double, y0(:,1) double, tn(1,1) double
+      h(1,1){mustBePositive}, p(1,1){mustBePositiveInteger}
+      yp(:,1) double=zeros(ceil(tn/h)+1,1);
+   end
+   alfn=ceil(alpha); m=ceil(tn/h)+1;
+   t=(0:(m-1))'*h; d1=length(y0); d2=length(alpha);
+   B=zeros(m,m,d2); g=double(genfunc(p));
+   for i=1:d2, w=get_vecw(alpha(i),m,g);
+      B(:,:,i)=rot90(hankel(w(end:-1:1)))/h^alpha(i);
+   end
+   c=y0./gamma(1:d1)'; du=zeros(m,d2);
+   u=0; for i=1:d1, u=u+c(i)*t.^(i-1); end
+   for i=1:d2
+      if alfn(i)==0, du(:,i)=u;
+      elseif alfn(i)<d1
+         for k=(alfn(i)+1):d1
+            du(:,i)=du(:,i)+(t.^(k-1-alpha(i)))*c(k)...
+                    *gamma(k)/gamma(k-alpha(i));
+   end, end, end
+   v=fsolve(f,yp,[],t,u,B,du); y=u+v;
+end