The main goal of this paper is to define a simple but effective method for approximating solutions of multi-order fractional differential equations relying on Caputo fractional derivative and under supplementary conditions. Our basis functions are based on some original generalization of the Bessel polynomials, which satisfy many properties shared by the classical orthogonal polynomials as given by Hermit, Laguerre, and Jacobi. The main advantages of our polynomials are two-fold: All the coefficients are positive and any collocation matrix of Bessel polynomials at positive points is strictly totally positive. By expanding the unknowns in a (truncated) series of basis functions at the collocation points, the solution of governing differential equation can be easily converted into the solution of a system of algebraic equations, thus reducing the computational complexities considerably. Several practical test problems also with some symmetries are given to show the validity and utility of the proposed technique. Comparisons with available exact solutions as well as with several alternative algorithms are also carried out. The main feature of our approach is the good performance both in terms of accuracy and simplicity for obtaining an approximation to the solution of differential equations of fractional order.

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In this research, numerical approximation to fractional Bagley‐Torvik equation as an important model arising in fluid mechanics is investigated. Our discretization algorithm is based on the local discontinuous Galerkin (LDG) schemes along with using the natural upwind fluxes, which enables us to solve the model problem element by element. This means that we require to solve a low‐order system of equations in each subinterval, hence avoiding the need for a full global solution. The proposed schemes are tested on a range of initial‐ and boundary‐value problems including a variable coefficient example, a nonsmooth problem, and some oscillatory test cases with exact solutions. Also, the validation of the proposed methods was compared with those obtained available existing computational procedures. Overall, it was found that LDG methods indicated highly satisfactory performance with comparatively lower degree of polynomials and number of elements compared with other numerical models.

The main purpose of this article is to investigate a novel set of (orthogonal) basis functions for treating a class of multi-order fractional pantograph differential equations (MOFPDEs) computationally. These polynomials, denoted by S n ( x ) and called special polynomials , were first discovered in a study of a certain family of isotropic turbulence fields. They are expressible in terms of the generalized Laguerre polynomials and are related to the Bessel and Srivastava–Singhal polynomials. Unlike the Laguerre polynomials, all coefficients of the special polynomials are positive. We further introduce the fractional order of the special polynomials and use them along with some suitable collocation points in a special matrix technique to treat fractional-order MOFPDEs. Moreover, the convergence analysis of these polynomials is established. Through five example applications, the utility and efficiency of the present matrix approach are demonstrated and comparisons with some existing numerical schemes have been performed in this class.

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The adjustable propagation length enhancement of the surface plasmon polariton (SPP) mode under the effects of the initial relative phase (ψ0) between interacting waves in difference frequency generation (DFG) based optical parametric amplification (OPA) are numerically considered. The waveguide is a silver coated PPLN planar waveguide. Obtained results indicate ultra long propagation length for the SPP mode could be achieved via manipulation of ψ0 in exact quasi phase matching (QPM) case up to 30 mm for initial pump intensity about 66 MW/cm for degenerate DFG (dDFG). For chirped QPM by mitigating the high depletion of the pump intensity, it is possible to enhance the SPP propagation length up to 43 mm for initial pump intensity about 135 MW/cm. In this case ψ0 does not affect the SPP propagation length except around a narrow range of unsuitable phases. The unsuitable phase is for exact QPM but is pump dependent for chirped QPM case. Using this unsuitable phase is the key parameter to the SPP propagation length enhancement via controlling ψ0. In this case with a high pump intensity, the pump and the SPP modes interact at longer distances which leads to the SPP propagation length enhancement.