init

FOTF Toolbox/nlfep.m

@@ -0,0 +1,40 @@
+function [y,t]=nlfep(fun,alpha,y0,tn,h,p,err)
+% nlfec - O(h^p) predictor solution of nonlinear multi-term FODEs
+%
+%    [y,t]=nlfep(fun,alpha,y0,tn,h,p,err)
+%
+%   fun - 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
+%   err - error tolerance
+%   y, t - the solution vector and time vector
+
+% Copyright (c) Dingyu Xue, Northeastern University, China
+% Last modified 30 March, 2017 
+% Last modified 18 May, 2022 
+   arguments
+      fun, alpha(1,:), y0(1,:), tn(1,1), h(1,1){mustBePositive}
+      p(1,1) {mustBePositiveInteger}, err(1,1) double=1e-10
+   end
+   m=ceil(tn/h)+1; t=(0:(m-1))'*h; ha=h.^alpha; z=0;
+   [T,dT,w,d2]=aux_func(t,y0,alpha,p); y=z+T(1); 
+   for k=1:m-1
+      zp=z(end); yp=zp+T(k+1); y=[y; yp]; z=[z; zp];
+      [zc,yc]=repeat_funs(fun,t,y,d2,w,k,z,ha,dT,T);
+      while abs(zc-zp)>err
+         yp=yc; zp=zc; y(end)=yp; z(end)=zp;
+         [zc,yc]=repeat_funs(fun,t,y,d2,w,k,z,ha,dT,T);
+   end, end
+% repeatedly called subfunction
+   function [zc,yc]=repeat_funs(fun,t,y,d2,w,k,z,ha,dT,T)
+      dy=zeros(1,d2-1);
+      for j=1:(d2-1)
+         dy(j)=w(1:k+1,j+1)'*z((k+1):-1:1)/ha(j+1)+dT(k,j+1);
+      end, f=fun(t(k+1),y(k+1),dy);
+      zc=((f-dT(k+1,1))*ha(1)-w(2:k+1,1)'*z(k:-1:1))/w(1,1); 
+      yc=zc+T(k+1);
+   end
+end