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Recently, I met a partial differential equation with integral term. At first, I wanted to solve it directly through NDSolve, but found that it was not feasible because MMA could not directly deal with partial differential equations with delay. $$ 2\frac{\partial ^2v}{\partial \theta ^2}-\frac{\partial w}{\partial \theta}+\frac{\partial ^3w}{\partial \theta ^3}=\frac{\partial ^2v}{\partial t^2} $$ $$ \frac{\partial v}{\partial \theta}-w-\frac{\partial ^3v}{\partial \theta ^3}-\frac{\partial ^4w}{\partial \theta ^4}=\frac{\partial ^2w}{\partial t^2}-q\left( \theta ,t \right) $$ $$ q\left( \theta ,t \right) =\exp \left( -t \right) -\exp \left[ -c\left( \theta \right) t \right] \int_{-r\left( \theta \right)}^t{\exp \left[ -c\left( \theta \right) x \right] \frac{\partial ^2w\left( \theta ,x \right)}{\partial x^2}dx} $$ $$ c\left( \theta \right) =\sqrt{1+8\sin ^2\left( \theta /2 \right)},r\left( \theta \right) =\sin ^2\left( \theta /2 \right) /750 $$ Where,$w$ and $v$ satisfy periodic boundary conditions, $ v\left( 0,t \right) =v\left( 2\pi ,t \right) ,w\left( 0,t \right) =w\left( 2\pi ,t \right)$, and initial conditions $ v\left( \theta ,0 \right) =0,w\left( \theta ,0 \right) =0,\frac{\partial v\left( \theta ,0 \right)}{\partial t}=0,\frac{\partial w\left( \theta ,0 \right)}{\partial t}=0 $.

Solution 1: So, I want to try whether MMA can solve delayed ordinary differential equations. Here, xzczd Trouble in a delay ordinary differential equation gives me a way to solve this equation by Laplace transform. However, when I used the same method to deal with partial differential equations, I encountered some problems and didn't know how to solve them.

r[θ_] = Sin[θ/2]^2/750; 
c[θ_] := Sqrt[1 + 8*Sin[θ/2]^2]; 
q[θ_, t_] := Exp[-t] - Exp[(-c[θ])*t]*Integrate[Exp[(-c[θ])*x]*D[w[θ, x], {x, 2}], {x, r[θ], t}]; 
pde = {2*D[v[θ, t], {θ, 2}] - D[w[θ, t], θ] + D[w[θ, t], {θ, 3}] == D[v[θ, t], {t, 2}], 
    D[v[θ, t], θ] - w[θ, t] - D[v[θ, t], {θ, 3}] - D[w[θ, t], {θ, 4}] == D[w[θ, t], {t, 2}] - q[θ, t]}; 
bc = {v[0, t] == v[2*Pi, t], w[0, t] == w[2*Pi, t]}; 
ic = {v[θ, 0] == 0, w[θ, 0] == 0, Derivative[0, 1][v][θ, 0] == 0, Derivative[0, 1][w][θ, 0] == 0}; 

enter image description here Solution 2: For such a problem, another method is to transform partial differential equations into ordinary differential equations by Method Of Lines https://reference.wolfram.com/language/tutorial/NDSolveMethodOfLines.html, and then solve it by Laplace transform. However, I don't know how to deal with the periodic boundary conditions of partial differential equations in the process of transformation. This part is not explicitly given in the help document, and can only be used in NDSovle. But NDSolve can't solve the delay differential equation. Therefore, the solution also encountered difficulties.

n = 24; 
Subscript[h, n] = 2*(Pi/n); 
r[θ_] = 2*Sin[θ/2]^2; 
c[θ_] := Sqrt[1 + 8*Sin[θ/2]^2]; 
q[θ_, t_] := Exp[-t] - Exp[(-c[θ])*t]*Integrate[Exp[(-c[θ])*x]*D[w[θ, x], {x, 2}], {x, r[θ], t}]; 
Q[t_] = Table[q[θ, t], {θ, 0, 2*Pi, Subscript[h, n]}]; 
V[t_] := Table[Subscript[v, i][t], {i, 0, n}]; 
W[t_] := Table[Subscript[w, i][t], {i, 0, n}]; 
bc0 = Subscript[v, 0][t] == Subscript[v, n][t]; 
bc0d = (D[#1, t] + sf*#1 & ) /@ bc0; 
bc1 = Subscript[w, 0][t] == Subscript[w, n][t]; 
bc1d = (D[#1, t] + sf*#1 & ) /@ bc1; 
dv1 = NDSolve`FiniteDifferenceDerivative[1, Subscript[h, n]*Range[0, n], V[t], PeriodicInterpolation -> True]; 
dv2 = NDSolve`FiniteDifferenceDerivative[2, Subscript[h, n]*Range[0, n], V[t], PeriodicInterpolation -> True]; 
dv3 = NDSolve`FiniteDifferenceDerivative[3, Subscript[h, n]*Range[0, n], V[t], PeriodicInterpolation -> True]; 
dw1 = NDSolve`FiniteDifferenceDerivative[1, Subscript[h, n]*Range[0, n], W[t], PeriodicInterpolation -> True]; 
dw2 = NDSolve`FiniteDifferenceDerivative[2, Subscript[h, n]*Range[0, n], W[t], PeriodicInterpolation -> True]; 
dw3 = NDSolve`FiniteDifferenceDerivative[3, Subscript[h, n]*Range[0, n], W[t], PeriodicInterpolation -> True]; 
dw4 = NDSolve`FiniteDifferenceDerivative[4, Subscript[h, n]*Range[0, n], W[t], PeriodicInterpolation -> True]; 
eq1 = Thread[2*dv2 - dw1 + dw3 == D[V[t], {t, 2}]]; 
eq2 = Thread[dv1 - W[t] + dv3 + dw4 == (D[W[t], {t, 2}] - Q[t])]; 
inc1 = Thread[V[0] == Table[0, {n + 1}]]; inc2 = Thread[D[V[t], t] == Table[0, {n + 1}]] /. t -> 0; 
inc3 = Thread[W[0] == Table[0, {n + 1}]]; inc4 = Thread[D[W[t], t] == Table[0, {n + 1}]] /. t -> 0; 
system = Join[eq1, eq2, inc1, inc2, inc3, inc4]; 
lines = NDSolve[system, Join[V[t], W[t]], {t, 0, 1}]

enter image description here Faced with this problem, I have made many attempts, but they are all difficult to overcome. I hope to get help here. Thank you very much for your suggestions on any of the above solutions.

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  • $\begingroup$ 1. You're using LaplaceTransform blindly, it cannot be used in $\theta$ direction in this case, for more info please check the property of LaplaceTransform carefully. 2. I'm afraid LaplaceTransform won't help much for this problem, just try Clear[r, c]; LaplaceTransform[q[\[Theta], t], t, s] 3. As to discretization, I'm afraid NDSolve won't help in this case. You probably need to discretize in $t$ direction, too. Please think carefully about what's happening inside Integrate. $\endgroup$ – xzczd Dec 15 '20 at 4:31
  • $\begingroup$ Your suggestion is very useful. Thank a lot 吧主. $\endgroup$ – mozeq Dec 15 '20 at 11:35
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    $\begingroup$ Another possibility for handling the integral is the iterative approach, the followings are some examples: mathematica.stackexchange.com/a/160620/1871 mathematica.stackexchange.com/a/148607/1871 mathematica.stackexchange.com/a/145303/1871 The method is expensive compared to discretization method, but maybe easier to implement. $\endgroup$ – xzczd Dec 15 '20 at 11:41

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