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The research reported in this paper deals with the existence of solutions of boundary value problem (BVP) of nonlinear neutral pantograph equations by means of complex fractional derivative in weighted spaces. The results were interpreted with the aid of classical fixed point theorems. The Ulam–Hyers–Rassias stability and Ulam–Hyers stability of differential equations are studied by utilizing the complex fractional derivative through the fixed point method.

In this work, the optimal control for a class of Hilfer fractional stochastic integrodifferential systems driven by Rosenblatt process and Poisson jumps has been discussed in infinite dimensional space involving the Hilfer fractional derivative. First, we study the existence and uniqueness of mild solution results are proved by the virtue of fractional calculus, successive approximation method and stochastic analysis techniques. Second, the optimal control of the proposed problem is presented by using Balder's theorem. Finally, an example is demonstrated to illustrate the obtained theoretical results.

This paper is devoted to investigate implicit differential equations with boundary condition, which involves the composite fractional derivative in weighted space. The existence and uniqueness of the solution are obtained using the classic fixed point theorems. As an application, an example is presented to illustrate the main results.

AbstractIn this paper, we discuss the existence, uniqueness and stability of implicit differential equation with Ψ-fractional derivative. The arguments are based upon Schauder fixed point theorem, Banach contraction principle and Ulam stability.

In this paper, we discuss the existence, uniqueness and stability of boundary value problem for differential equation with Hilfer-Katugampola fractional derivative. The arguments are based upon Schaefer’s fixed point theorem, Banach contraction principle and Ulam type stability.

In this paper, we investigate the existence of solution of integro-differential equations (IDEs) with Hilfer-Hadamard fractional derivative. The main results are obtained by using Schaefer’s fixed point theorem. Some Ulam stability results are presented.

In this paper we discuses the periodic solutions of particular cases of the following general system of difference equations xn+1=a1+a2yna3zn+a4xn-1zn,yn+1=b1zn-1+b2znb3xnyn+b4xnyn-1,zn+1=c1zn-1+c2znc3xn-1yn-1+c4xn-1yn+c5xnyn,$$x_{n + 1} = {{a_1 + a_2 y_n } \over {a_3 z_n + a_4 x_{n - 1} z_n }},y_{n + 1} = {{b_1 z_{n - 1} + b_2 z_n } \over {b_3 x_n y_n + b_4 x_n y_{n - 1} }},z_{n + 1} = {{c_1 z_{n - 1} + c_2 z_n } \over {c_3 x_{n - 1} y_{n - 1} + c_4 x_{n - 1} y_n + c_5 x_n y_n }},$$where the initial conditions x−1, x0, y−1, y0, z−1, and z0are arbitrary nonzero real numbers and ai, bi, and cj, for i= 1, 2, 3, 4, j= 1, 2, 3, 4, 5 are non-negative real numbers. The results obtained are new and contained as special cases some other results.

The intensity of light scattered by a coated sphere illuminated with an off-axis Gaussian beam is calculated. Results are shown for different beam positions with respect to the sphere. As the beam is shifted further away from the surface of the sphere, the higher-Q morphology-dependent resonances become increasingly important in the backscatter spectra, and the angular scattering intensity becomes smoother.The scattered intensity depends on the beam position, the refractive indices of the core and coat, the radius of the core, and the thickness of the coat. As the beam is moved further away from the sphere, the effect of the core on the scattering intensity decreases. When the incident Gaussian beam is focused outside of a particle with a relatively small core, the scattering spectra and angular scattering patterns become similar to those of a homogeneous sphere having the refractive index of the coat. These calculated results suggest that measurements of spectral scattering and angular scattering patterns for several Gaussian beam positions could be useful for the characterization of coated spheres.

The electric energy in a lossless or lossy spherical particle that is illuminated with a plane wave or a Gaussian beam is investigated. The analysis uses a combination of the plane-wave spectrum technique and the T-matrix method. Expressions for the electric energy in any mode as well as the total electric energy inside the particle are given. The amount of energy coupling into the particle for different beam illuminations is also investigated. The high-Q (low-order) resonant modes can dominate the electric energy inside a spherical particle many linewidths away from the resonance location, particularly if the beam is focused at the droplet edge or outside the droplet. If the sphere is lossy, low-order modes can still dominate the electric energy if the beam is focused far enough outside the sphere. As the absorption coefficient of the particle increases, the energy in a high-Q mode decreases much faster at the resonance frequency than it does at near or off-resonance frequencies. The effects of the absorption on the dominance of the internal fields by a high-Q mode decreases as the beam is shifted farther away from the particle. As the beam is shifted farther away from the particle the fraction of the incident energy coupled into the sphere at resonance first increases and then decreases. Although the coupled energy decreases as the beam is shifted farther from the sphere, most of that energy is in the lowest-order mode.