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seen Dec 29 '13 at 18:26

ClearAll["Global`*"];

T = 100; M = 100000; h = T/M; [Alpha]1 = 1; [Alpha]2 = 0.75; x[0] = 10; y[0] = 0.01; [Epsilon] = 0.01;

f1[x_, y_] := y; f2[x_, y_] := -x - [Epsilon]*(y + y^(2/3));

Do[b1[k] = k^[Alpha]1 - (k - 1)^[Alpha]1; a1[k] = (k + 1)^([Alpha]1 + 1) - 2*k^([Alpha]1 + 1) + (k - 1)^([Alpha]1 + 1), {k, 1, M}]

Do[b2[k] = k^[Alpha]2 - (k - 1)^[Alpha]2; a2[k] = (k + 1)^([Alpha]2 + 1) - 2*k^([Alpha]2 + 1) + (k - 1)^([Alpha]2 + 1), {k, 1, M}]

Do[p1 = x[0] + h^[Alpha]1/Gamma[[Alpha]1 + 1]* Sum[b1[j - k]f1[x[k], y[k]], {k, 0, j - 1}]; p2 = y[0] + h^[Alpha]2/Gamma[[Alpha]2 + 1] Sum[b2[j - k]f2[x[k], y[k]], {k, 0, j - 1}]; x[j] = x[0] + h^[Alpha]1/ Gamma[[Alpha]1 + 2](f1[p1, p2] + ((j - 1)^([Alpha]1 + 1) - (j - 1 - [Alpha]1)* j^[Alpha]1)*f1[x[0], y[0]] + Sum[a1[j - k]f1[x[k], y[k]], {k, 1, j - 1}]); y[j] = y[0] + h^[Alpha]2/ Gamma[[Alpha]2 + 2](f2[p1, p2] + ((j - 1)^([Alpha]2 + 1) - (j - 1 - [Alpha]2)* j^[Alpha]2)*f2[x[0], y[0]] + Sum[a2[j - k]*f2[x[k], y[k]], {k, 1, j - 1}]), {j, 1, M}]

ListLinePlot[Table[{m*h, Re[x[m]]}, {m, 0, M}], Axes -> {False, True}, Frame -> True, PlotRange -> {{0, 100}, {-10, 20}}, FrameLabel -> {"t", "x(t)"}, PlotRange -> {{0, 100}, {-10, 20}}, FrameTicks -> {{{-10, 0, 10, 20}, None}, {{0, 20, 40, 60, 80, 100}, None}}, RotateLabel -> False, PlotStyle -> Black]

ListLinePlot[Table[{m*h, Re[y[m]]}, {m, 0, M}], Axes -> {False, True}, Frame -> True, PlotRange -> {{0, 100}, {-10, 10}}, FrameLabel -> {"t", "y(t)"}, PlotRange -> {{0, 100}, {-10, 10}}, FrameTicks -> {{{-10, 0, 10}, None}, {{0, 20, 40, 60, 80, 100}, None}}, RotateLabel -> False, PlotStyle -> Black]

ListLinePlot[Table[{Re[x[m]], Re[y[m]]}, {m, 0, M}], PlotRange -> All, Frame -> True, Axes -> False, FrameLabel -> {"x(t)", "y(t)"}, PlotRange -> {{-10, 10}, {-10, 10}}, FrameTicks -> {{{-10, 0, 10}, None}, {{-10, -5, 0, 5, 10}, None}}, RotateLabel -> False, PlotStyle -> Black]


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