how can I use summation and index for the series

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work wolf 2022-5-24

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https://ww2.mathworks.cn/matlabcentral/answers/1725685-how-can-i-use-summation-and-index-for-the-series

评论： Torsten 2022-5-25

在 MATLAB Online 中打开

how can I use summation and index for the following series by using matlab

Edit:

the pervious Equation solution for

My trying to write the following code, but i don't know if it is right ot no

u0=0;
alpha=0.5;%[0.1 0.3 0.5 0.6 0.66 0.9 1]; matrix 
a=0; b=1;n=100;
h=(b-a)/n;
t=a:h:b;
f=@(t,u) (-u.^4) +  ( gamma(2*alpha+1)./gamma(alpha+1) ).*(t.^alpha) - ( 2./gamma(3-alpha) ).*(t.^(2-alpha)) + (t.^(2*alpha)-t.^2).^4;
exact=@(t) t.^(2*alpha)-t.^2;
%-----a'ks---------------------------------------
a_kth=@(k) (k+1).^(1-alpha)-(k).^(1-alpha);
u_p=zeros(1,n); z_p=zeros(1,n); u=zeros(1,n);
s=0;
for i=1
    u_p(i)=a_kth(i-1)*u0;    %s=s+q^keval(subs(fun,xq^k));
    z_p(i)=gamma(2-alpha)*( h.^alpha ) * f(t(i),u_p(i));
    u(i)=u_p(i)+gamma(2-alpha)*( h.^alpha ) * f(t(i),u_p(i)+z_p(i));
end
%--------------------------------------
for i=2:n
for k=1:i-1
    s=s+(a_kth(i-1-k)-a_kth(i-k))*u(k);
    u_p(i)=a_kth(i-1)*u0+s;    %s=s+q^keval(subs(fun,xq^k));
    z_p(i)=gamma(2-alpha)*( h.^alpha ) * f(t(i),u_p(i));
    u(i)=u_p(i)+gamma(2-alpha)*( h.^alpha ) * f(t(i),u_p(i)+z_p(i));
end
end
u1=[ u0 u];
plot(t,u1)
exact1=exact(t);
error=exact1-u1;

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work wolf 2022-5-25

编辑：work wolf 2022-5-25

在 MATLAB Online 中打开

@Torsten yes. f depends on u_n , where u_n denotes the approximate value of u(t) at t= t_n .

Note: i am trying to write the folowing code , but i don't know if it right or wrong?!

u0=0;
alpha=0.5;%[0.1 0.3 0.5 0.6 0.66 0.9 1]; matrix 
a=0; b=1;n=100;
h=(b-a)/n;
t=a:h:b;
f=@(t,u) (-u.^4) +  ( gamma(2*alpha+1)./gamma(alpha+1) ).*(t.^alpha) - ( 2./gamma(3-alpha) ).*(t.^(2-alpha)) + (t.^(2*alpha)-t.^2).^4;
exact=@(t) t.^(2*alpha)-t.^2;
%-----a'ks---------------------------------------
a_kth=@(k) (k+1).^(1-alpha)-(k).^(1-alpha);
u_p=zeros(1,n); z_p=zeros(1,n); u=zeros(1,n);
s=0;
for i=1
    u_p(i)=a_kth(i-1)*u0;    %s=s+q^keval(subs(fun,xq^k));
    z_p(i)=gamma(2-alpha)*( h.^alpha ) * f(t(i),u_p(i));
    u(i)=u_p(i)+gamma(2-alpha)*( h.^alpha ) * f(t(i),u_p(i)+z_p(i));
end
%--------------------------------------
for i=2:n
    for k=1:i-1
        s=s+(a_kth(i-1-k)-a_kth(i-k))*u(k);
        u_p(i)=a_kth(i-1)*u0+s;    %s=s+q^keval(subs(fun,xq^k));
        z_p(i)=gamma(2-alpha)*( h.^alpha ) * f(t(i),u_p(i));
        u(i)=u_p(i)+gamma(2-alpha)*( h.^alpha ) * f(t(i),u_p(i)+z_p(i));
    end
end
u1=[ u0 u];
plot(t,u1)
exact1=exact(t);
error=exact1-u1;