p[x_,y_]:=Sin[x]/(Cosh[y]-Cos[x])

{Plot3D[p[x,y],{x,0,Pi},{y,-5,5},PlotRange->{0,5}],Plot3D[p[Pi-x,y],{x,0,Pi},{y,-5,5},PlotRange->{0,5}]}

q1[x_,y_]:=(Sin[x])^2/(y^2+1-Cos[x])

{Plot3D[q1[x,y],{x,0,Pi},{y,-5,5},PlotRange->{0,2}],Plot3D[q1[Pi-x,y],{x,0,Pi},{y,-5,5},PlotRange->{0,2}]}

Limit[q1[x,0],x->0]

q2[x_,y_]:=(Sin[x])^2/(Cosh[y]-Cos[x])

{Plot3D[q2[x,y],{x,0,Pi},{y,-5,5},PlotRange->All],Plot3D[q2[Pi-x,y],{x,0,Pi},{y,-5,5},PlotRange->All]}

Limit[q2[x,0],x->0]

q3[x_,y_]:=(Sin[x]/(Cosh[y]-Cos[x]))^(1/2)

{Plot3D[q3[x,y],{x,0,Pi},{y,-5,5},PlotRange->{0,3}],Plot3D[q3[Pi-x,y],{x,0,Pi},{y,-5,5},PlotRange->{0,3}]}

Limit[q3[x,0],x->0]

intp[x_]:=Integrate[q2[x,y],{y,-Infinity,Infinity}]

FullSimplify[intp[x],Assumptions->0<x<Pi]

sL[t_]:=DiracDelta[t-t0]+DiracDelta[t-(t0+7)];
sR[t_]:=DiracDelta[t-t1]+DiracDelta[t-(t1+5)];

sx[x_,y_]:=1/(2 Pi) Integrate[q3[x,t-y] sL[t],{t,-Infinity,Infinity},Assumptions->{t0 ∈ Reals,Pi>x>0}]+1/(2 Pi) Integrate[q3[Pi-x,t-y] sR[t],{t,-Infinity,Infinity},Assumptions->{t1 ∈ Reals,Pi>x>0}]

sx[x,y]

sx[x_,y_,t0_,t1_]:=(((Sqrt[((Sin[x])/(-Cos[x]+Cosh[t0-y]))])+(Sqrt[((Sin[x])/(-Cos[x]+Cosh[7+t0-y]))]))/(2 π))+(((Sqrt[((Sin[x])/(Cos[x]+Cosh[t1-y]))])+(Sqrt[((Sin[x])/(Cos[x]+Cosh[5+t1-y]))]))/(2 π))

Plot3D[sx[x,y,5,6],{x,0,Pi},{y,0,20},PlotRange->{0,1},MaxRecursion->5]

sx[x_,y_]:=1/(2 Pi) Integrate[q2[x,t-y] sL[t],{t,-Infinity,Infinity},Assumptions->{t0 ∈ Reals,Pi>x>0}]+1/(2 Pi) Integrate[q2[Pi-x,t-y] sR[t],{t,-Infinity,Infinity},Assumptions->{t1 ∈ Reals,Pi>x>0}]

sx[x,y]

sx[x_,y_,t0_,t1_]:=((((1/(-Cos[x]+Cosh[t0-y]))+(1/(-Cos[x]+Cosh[7+t0-y]))) ((Sin[x])^2))/(2 π))+((((1/(Cos[x]+Cosh[t1-y]))+(1/(Cos[x]+Cosh[5+t1-y]))) ((Sin[x])^2))/(2 π))

Plot3D[sx[x,y,5,6],{x,0,Pi},{y,0,20},PlotRange->All,MaxRecursion->5,PlotPoints->100,ImageSize->800]

the above is figure poiss3.png and has the analytic form gL[x] = gR[x] = Sin[x]

g[x_]:=x;
sxG[x_,y_]:=(Pi-g[x])/(2 Pi) Integrate[ q2[x,t-y] sL[t],{t,-Infinity,Infinity},Assumptions->{t0∈Reals,Pi>x>0}]+g[x]/(2 Pi) Integrate[q2[Pi-x,t-y] sR[t],{t,-Infinity,Infinity},Assumptions->{t1∈Reals,Pi>x>0}]

sxG[x,y]

sxG[x_,y_,t0_,t1_]:=((π-x) (1/(-Cos[x]+Cosh[t0-y])+1/(-Cos[x]+Cosh[7+t0-y])) Sin[x]^2)/(2 π)+(x (1/(Cos[x]+Cosh[t1-y])+1/(Cos[x]+Cosh[5-t1+y])) Sin[x]^2)/(2 π);

Plot3D[sxG[x,y,5,11],{x,0,Pi},{y,0,20},PlotRange->All,MaxRecursion->5,PlotPoints->100,ImageSize->800]

the above figure is poiss2.png and has the same analytical form as case immediately to follow

gL[x_]:=(Pi-x) Sin[x];
gR[x_]:=x Sin[x];
sxG[x_,y_]:=gL[x]/(2 Pi) Integrate[p[x,t-y] sL[t],{t,-Infinity,Infinity},Assumptions->{t0∈Reals,Pi>x>0}]+gR[x]/(2 Pi) Integrate[p[Pi-x,t-y] sR[t],{t,-Infinity,Infinity},Assumptions->{t1∈Reals,Pi>x>0}]

sxG[x,y]

sx[x_,y_]:=1/(2 Pi) Integrate[q1[x,t-y] sL[t],{t,-Infinity,Infinity},Assumptions->{t0 ∈ Reals,Pi>x>0}]+1/(2 Pi) Integrate[q1[Pi-x,t-y] sR[t],{t,-Infinity,Infinity},Assumptions->{t1 ∈ Reals,Pi>x>0}]

sx[x,y]

sx[x_,y_,t0_,t1_]:=(1/(2 π))If[y-(Sqrt[-1-Cos[x]])∉Reals&&y+(Sqrt[-1-Cos[x]])∉Reals,((1/(1+(((t1-y))^2)+Cos[x]))+(1/(26+10 t1+(t1^2)-2 (5+t1) y+(y^2)+Cos[x]))) ((Sin[x])^2),Integrate[((DiracDelta[t-t1] ((Sin[x])^2))/(1+(((t-y))^2)+Cos[x]))+((DiracDelta[5-t+t1] ((Sin[x])^2))/(1+(((t-y))^2)+Cos[x])),{t,-∞,∞},Assumptions->x<π&&x>0&&(y+(Sqrt[-1-Cos[x]])∈Reals||y-(Sqrt[-1-Cos[x]])∈Reals)&&t1∈Reals]]+(1/(2 π))If[y-(Sqrt[-1+Cos[x]])∉Reals&&y+(Sqrt[-1+Cos[x]])∉Reals,((1/(1+(((t0-y))^2)-Cos[x]))+(1/(50+14 t0+(t0^2)-2 (7+t0) y+(y^2)-Cos[x]))) ((Sin[x])^2),Integrate[((DiracDelta[t-t0] ((Sin[x])^2))/(1+(((t-y))^2)-Cos[x]))+((DiracDelta[7-t+t0] ((Sin[x])^2))/(1+(((t-y))^2)-Cos[x])),{t,-∞,∞},Assumptions->x<π&&x>0&&(y+(Sqrt[-1+Cos[x]])∈Reals||y-(Sqrt[-1+Cos[x]])∈Reals)&&t0∈Reals]]

Plot3D[sx[x,y,5,6],{x,0,Pi},{y,0,20},PlotRange->All,MaxRecursion->5,PlotPoints->100]