I am trying to solve the following integro-differential equation: $$ a(\varepsilon) f'(\varepsilon) + b(\varepsilon) f(\varepsilon) + \int_{\varepsilon}^{\varepsilon + \varepsilon_1} R_1(\epsilon) f(\epsilon) d\epsilon - \int_{\varepsilon - \varepsilon_2}^{\varepsilon} R_2(\epsilon) f(\epsilon) d\epsilon = 0, $$ where $\varepsilon_1 = 10$, $\varepsilon_2 = 20$.

Note that the reduction of this equation into a second-order ODE is possible, but is much harder to solve. See, e.g., this question.

Taking

eps1 = 10;
eps2 = 20;
epsmin = .025;
epsmax = 150;
a[eps_] := 1 + eps
b[eps_] := eps^1.4
R1[eps_] := Piecewise[{{eps, eps1 <= eps <= epsmax}|}, 0]
R2[eps_] := Piecewise[{{eps, eps2 <= eps <= epsmax}|}, 0]

We know that f[eps] > 0 for epsmin <= eps <= epsmax, and that f[eps] = 0 for eps <= epsmin and eps >= epsmax. For eps >= epsmax, all derivatives of f[eps] also become 0. Also,

Integrate[Sqrt[eps] f[eps], {eps, epsmin, epsmax}] == 1;

I have tried to use NDSolve with NIntegrate as follows:

solBW = NDSolve[{f'[eps]+b[eps] f[eps]/a[eps] + 1/a[eps] NIntegrate[R1[eps1] f[eps1],{eps1, eps, eps + eps1}] - 1/a[eps] NIntegrate[R2[eps1] f[eps1],{eps1, eps - eps2, eps}], f[epsmax]==0}, f, {eps, epsmin, epsmin}];

but it returns

NIntegrate: eps1 = eps is not a valid limit of integration. 

Any hint on how to solve this equation numerically is appreciated.

I am trying to solve the following integro-differential equation: $$ a(\varepsilon) f'(\varepsilon) + b(\varepsilon) f(\varepsilon) + \int_{\varepsilon}^{\varepsilon + \varepsilon_1} R_1(\epsilon) f(\epsilon) d\epsilon - \int_{\varepsilon - \varepsilon_2}^{\varepsilon} R_2(\epsilon) f(\epsilon) d\epsilon = 0, $$ where $\varepsilon_1 = 10$, $\varepsilon_2 = 20$.

Note that the reduction of this equation into a second-order ODE is possible, but is much harder to solve. See, e.g., this question.

Taking

eps1 = 10;
eps2 = 20;
epsmin = .025;
epsmax = 150;
a[eps_] := 1 + eps
b[eps_] := eps^1.4
R1[eps_] := Piecewise[{{eps, eps1 <= eps <= epsmax}}, 0]
R2[eps_] := Piecewise[{{eps, eps2 <= eps <= epsmax}}, 0]

We know that f[eps] > 0 for epsmin <= eps <= epsmax, and that f[eps] = 0 for eps <= epsmin and eps >= epsmax. For eps >= epsmax, all derivatives of f[eps] also become 0. Also,

Integrate[Sqrt[eps] f[eps], {eps, epsmin, epsmax}] == 1;

I have tried to use NDSolve with NIntegrate as follows:

solBW = NDSolve[{a[eps]f'[eps] + b[eps]f[eps] + NIntegrate[R1[y]f[y],{y,eps,eps + eps1}] - NIntegrate[R2[y]f[y],{y, eps - eps2, eps}] == 0, f[epsmax] == 0}, f, {eps, epsmax, epsmin}];

but it returns

NIntegrate: y = eps is not a valid limit of integration. 

Any hint on how to solve this equation numerically is appreciated.

I am trying to solve the following integro-differential equation: $$ a(\varepsilon) f'(\varepsilon) + b(\varepsilon) f(\varepsilon) + \int_{\varepsilon}^{\varepsilon + \varepsilon_1} R_1(\epsilon) f(\epsilon) d\epsilon - \int_{\varepsilon - \varepsilon_2}^{\varepsilon} R_2(\epsilon) f(\epsilon) d\epsilon = 0, $$ where $\varepsilon_1 = 10$, $\varepsilon_2 = 20$.

Note that the reduction of this equation into a second-order ODE is possible, but is much harder to solve. See, e.g., this question.

Taking

eps1 = 10;
eps2 = 20;
epsmin = .025;
epsmax = 150;
a[eps_] := 1 + eps
b[eps_] := eps^1.4
R1[eps_] := Piecewise[{{eps, eps1 <= eps <= epsmax}|}, 0]
R2[eps_] := Piecewise[{{eps, eps2 <= eps <= epsmax}|}, 0]

We know that f[eps] > 0 for epsmin <= eps <= epsmax, and that f[eps] = 0 for eps <= epsmin and eps >= epsmax. For eps >= epsmax, all derivatives of f[eps] also become 0. Also,

Integrate[Sqrt[eps] f[eps], {eps, epsmin, epsmax}] == 1;

Any hint on how to solve this equation numerically is appreciated.

I am trying to solve the following integro-differential equation: $$ a(\varepsilon) f'(\varepsilon) + b(\varepsilon) f(\varepsilon) + \int_{\varepsilon}^{\varepsilon + \varepsilon_1} R_1(\epsilon) f(\epsilon) d\epsilon - \int_{\varepsilon - \varepsilon_2}^{\varepsilon} R_2(\epsilon) f(\epsilon) d\epsilon = 0, $$ where $\varepsilon_1 = 10$, $\varepsilon_2 = 20$.

Note that the reduction of this equation into a second-order ODE is possible, but is much harder to solve. See, e.g., this question.

Taking

eps1 = 10;
eps2 = 20;
epsmin = .025;
epsmax = 150;
a[eps_] := 1 + eps
b[eps_] := eps^1.4
R1[eps_] := Piecewise[{{eps, eps1 <= eps <= epsmax}|}, 0]
R2[eps_] := Piecewise[{{eps, eps2 <= eps <= epsmax}|}, 0]

We know that f[eps] > 0 for epsmin <= eps <= epsmax, and that f[eps] = 0 for eps <= epsmin and eps >= epsmax. For eps >= epsmax, all derivatives of f[eps] also become 0. Also,

Integrate[Sqrt[eps] f[eps], {eps, epsmin, epsmax}] == 1;

I have tried to use NDSolve with NIntegrate as follows:

solBW = NDSolve[{f'[eps]+b[eps] f[eps]/a[eps] + 1/a[eps] NIntegrate[R1[eps1] f[eps1],{eps1, eps, eps + eps1}] - 1/a[eps] NIntegrate[R2[eps1] f[eps1],{eps1, eps - eps2, eps}], f[epsmax]==0}, f, {eps, epsmin, epsmin}];

but it returns

NIntegrate: eps1 = eps is not a valid limit of integration. 

Any hint on how to solve this equation numerically is appreciated.