Two-dimensional wavelets collocation method for electromagnetic waves in dielectric media

This work presents an alternative solution for the mathematical analysis of the fractional waves in dielectric media. For the fractional wave equation, the Caputo fractional derivative was considered, the order of the spatial and temporal fractional derivatives are , respectively. In this analysis, we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space and time derivatives into the fractional wave equation. We will consider source free Maxwell equations in isotropic and homogeneous dielectric medium. The general solutions obtained in our research have been expressed in terms of the multivariate Mittag–Leffler functions, these functions depend only on the parameters and preserving the appropriated physical units according to the system studied.

The generation and detection of scalar or pseudoscalar high-frequency gravitational waves in some dielectrics, excited by ultrashort intense laser pulses are proposed. Hidden photons and axion particles are elemental excitations of such gravitational waves. The properties of longitudinal electromagnetic waves in dielectric media have been described from the point of view of photon-hidden photon conversion processes in these crystals. Scalar bound photonic states in dielectric media are analyzed. The experimental schemes for photon-hidden photon conversion with taking part in longitudinal electromagnetic excitations in media have been proposed. Hertz gravitational laboratory experiments with the illumination of the dielectric media by intense ultrashort laser pulses and the “Light Sching Trough Wall” scheme using are analyzed.

The propagation of electromagnetic waves in dielectric medium with decreasing perturbations (defects) is studied. We use the Maxwell operator M=∇×p(x)∇× and the perturbed operator Mλ=∇×(p(x)+λq(x))∇× to describe the “background” medium and perturbed medium, respectively, where λ>0 is the coupling constant. Under the assumption that M has a spectral gap G, we prove that a defect can create an eigenvalue of any given fixed value τ∊G for some λ>0, provided that the support of the defect is large enough. We also consider the asymptotic distribution of eigenvalues in the gap and bounds on the number of eigenvalue branches.

Maxwell’s equations consist of four first-order partial differential equations. First we deal with basic properties of Maxwell’s equations. Next we show how equations of electromagnetic wave motion are derived from Maxwell’s equations along vector analysis. It is important to realize that the generation of the electromagnetic wave is a direct consequence of the interplay between the electric field and magnetic field that both change with time. We deal with behaviors of electromagnetic waves in dielectric media where no true charge exists. At a first glance, this restriction seems to narrow a range of application of principles of electromagnetism. In practice, however, such a situation is universalistic; topics cover a wide range of electromagnetic phenomena, e.g., light propagation in dielectrics including water, glass, polymers. Polarized properties characterize the electromagnetic waves. These include linear, circular, and elliptic polarizations. The characteristics are important both from a fundamental aspect and from the point of view of optical applications.

Local nonlinear electrodynamics

The principal aim of this study is to present a new algorithm for solving some kinds of weakly singular fractional integro-differential equations. The suggested algorithm uses the shifted sixthkind Chebyshev polynomials together with the collocation method. Using the suggested algorithm and resultant operational matrices, the main equation converts into a system of algebraic equations which can be efficiently solved. Some theorems are proved and used to deduce an upper error bound for this method. Also, several examples are presented to illustrate the efficiency of the suggested algorithm compared to other methods in the literature. The suggested algorithm provides accurate results, even using a few terms of the proposed expansion.

Generally, fractional partial integro-differential equations (FPIDEs) play a vital role in modeling various complex phenomena. Because of the several applications of FPIDEs in applied sciences, mathematicians have taken a keen interest in developing and utilizing the various techniques for its solutions. In this context, the exact and analytical solutions are not very easy to investigate the solution of FPIDEs. In this article, a novel analytical approach that is known as the Laplace adomian decomposition method is implemented to calculate the solutions of FPIDEs. We obtain the approximate solution of the nonlinear FPIDEs. The results are discussed using graphs and tables. The graphs and tables have shown the greater accuracy of the suggested method compared to the extended cubic-B splice method. The accuracy of the suggested method is higher at all fractional orders of the derivatives. A sufficient degree of accuracy is achieved with fewer calculations with a simple procedure. The presented method requires no parametrization or discretization and, therefore, can be extended for the solutions of other nonlinear FPIDEs and their systems.

The differential equation plays an important role in the mathematical modeling of many physical phenomena that exists in the field of science and engineering.In the present paper, the numerical solution of nonlinear initial and boundary value problems is obtained by using Chebyshev Wavelet Collocation Method (CWCM) and Haar Wavelet Collocation Method (HWCM).The nonlinear equation is linearised using the quasilinearization technique and then the wavelet collocation methods are applied to linearised differential equations to convert it into a system of algebraic equations which can be handled easily.The solution is obtained at the specified points using Lagrange's interpolation technique.The results obtained by wavelet collocation methods are compared with the exact solutions.The tables and graphs show the efficiency of the wavelet collocation methods and we analyze that the method is accurate for a smaller number of collocation points.Thus, the comparative study highlights the advantages of wavelet collocation methods.

In this paper, by using the collocation method the operational matrix of Caputo fractional derivative was constructed for the cubic B-spline scaling function and wavelets. We apply the operational matrix and the Collocation method to solve linear fractional Volterra integro-differential equations. By using the principal characteristic of this technique, we have the operational matrix, which is applied to solve any linear fractional Volterra integro-differential equation by reducing the time, computer memory occupation and convert to a system of linear equations. Finally, the validity and applicability of the new technique will be shown by some numerical examples and convergence analysis.

A fast numerical method for fractional partial integro-differential equations with spatial-time delays

In this paper, we study a class of nonlinear time fractional partial integro-differential equations. The main results for the problem are obtained by the measure of noncompactness, solution operator, convex-power condensing operator and general Banach contraction mapping principle.

In this paper, we use operational matrices of Chebyshev polynomials to solve fractional partial differential equations (FPDEs). We approximate the second partial derivative of the solution of linear FPDEs by operational matrices of shifted Chebyshev polynomials. We apply the operational matrix of integration and fractional integration to obtain approximations of (fractional) partial derivatives of the solution and the approximation of the solution. Then we substitute the operational matrix approximations in the FPDEs to obtain a system of linear algebraic equations. Finally, solving this system, we obtain the approximate solution. Numerical experiments show an exponential rate of convergence and hence the efficiency and effectiveness of the method.

A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The convergence analysis of Bell wavelets approximation is discussed. The fractional integro-differential equations can be simplified to a system of algebraic equations by using a truncated Bell wavelets series and the fractional operational matrix. The proposed method’s efficacy is supported via various examples.

Spectral element technique for nonlinear fractional evolution equation, stability and convergence analysis

In this article, we consider a new class of fractional impulsive neutral stochastic functional integro-differential equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, fractional calculus, analytic α-resolvent operator and suitable fixed point theorems, we prove the existence of mild solutions and optimal mild solutions for these equations. Second, the existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. The results are obtained under weaker conditions in the sense of the fractional power arguments. Finally, an example is given for demonstration.