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I am trying to plot u(t) and v(t) vs t but no luck. I have also used RecurrenceTable but of no use. Here are the couple functions

u[0] = 0.5; alpha = 1/2; t1 = 2; l = 20; v[0] = 0.5;

u[t_] := Piecewise[{{u[0] +
NIntegrate[(u[t/2] + Cos[v[t/2]])/(t^3 + 100), {t, 0, t1}],
0 <= x <= t1}, {u[1] +
NIntegrate[(u[t/2] + 1/Sqrt[alpha]*Cos[v[t/2]])/(t^3 + 100), {t,
t1, t}], t1 <= t <= l}}]

v[t_] := Piecewise[{{v[0] +
NIntegrate[(v[t/2] + Sin[v[t/2]])/(t^5 + 100), {t, 0, t1}],
0 <= t <= t1}, {v[1] +
1/Sqrt[alpha]*
NIntegrate[(v[t/2] + Sin[v[t/2]])/(t^5 + 100), {t, t1, t}],
t1 <= t <= l}}]

My try

vTable = RecurrenceTable[{v[t] == Piecewise[{{v[0] + 
         NIntegrate[(Sin[v[t/2]])/(x^5 + 100), {x, 0, t1}], 
        0 <= t <= t1}, {v[1] + 
         1/Sqrt[alpha]*
          NIntegrate[(Sin[v[t/2]])/(x^5 + 100), {x, t1, t}], 
        t1 <= t <= l}}], v[0] == 0.5}, v, {t, 0, l}];

uTable = RecurrenceTable[{u[t] == Piecewise[{{u[0] + 
         NIntegrate[(u[t/2] + Cos[vTable[[t]]])/(x^3 + 100), {x, 0, t1}], 
        0 <= t <= t1}, {u[1] + 
         1/Sqrt[alpha]*
          NIntegrate[(u[t/2] + 1/Sqrt[alpha]*Cos[vTable[[t]]])/(x^3 + 100), {x, 
           t1, t}], t1 <= t <= l}}],u[0] == 0.5}, u, {t, 0, l}];
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  • 1
    $\begingroup$ Are your 2 functions coupled together? Could you simplify the question by just asking about 1 of them? Simpler questions get more attention. $\endgroup$
    – mikado
    Jan 15, 2023 at 11:05
  • 1
    $\begingroup$ First part in your Piecewisedefinitions should be 0 <= t<= t1(not 0 <= x <= t1) I think $\endgroup$ Jan 15, 2023 at 12:16

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