The expression and properties of Green’s function for a class of nonlinear fractional differential equations with integral boundary conditions are studied and employed to obtain some results on the existence of positive solutions by using fixed point theorem in cones. The proofs are based upon the reduction of problem considered to the equivalent Fredholm integral equation of second kind. The r...

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green’s formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for t...

in this paper, an effective direct method to determine the numerical solution of linear and nonlinear fredholm and volterra integral and integro-differential equations is proposed. the method is based on expanding the required approximate solution as the elements of chebyshev cardinal functions. the operational matrices for the integration and product of the chebyshev cardinal functions are des...

The optical eld in a weakly guiding homogeneous waveguide satis es scalar Helmholtz equations in both the core and cladding, and transmission conditions on the boundary. The transverse wavenumbers of the two scalar Helmholtz equations for the core and cladding are di erent and both of them depend on the propagation constants in the longitudinal direction of the waveguide. Two di erent systems o...