Fractional Series Expansion Research Articles

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

Simultaneous Development of Shocks and Cusps for 2D Euler with Azimuthal Symmetry from Smooth Data

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called $$C^{\frac{1}{3}} $$ pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau & Lifschitz [12, Chapter IX, §96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have $$C^{1, {\frac{1}{2}} }$$ one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form $$C^{1, {\frac{1}{2}} }$$ one-sided cusps while the pressure and normal velocity remain $$C^2$$ ; as such, we term this surface a weak contact discontinuity.

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

Many phenomena in physics and engineering can be built by linear and nonlinear fractional partial differential equations which are considered an accurate instrument to interpret these phenomena. In the current manuscript, the approximate analytical solutions for linear and nonlinear time-fractional Swift-Hohenberg equations are created and studied by means of a recent superb technique, named the Laplace residual power series (LRPS) technique under the time-Caputo fractional derivatives. The proposed technique is a combination of the generalized Taylor’s formula and the Laplace transform operator, which depends mainly on the concept of limit at infinity to find the unknown functions for the fractional series expansions in the Laplace space with fewer computations and more accuracy comparing with the classical RPS that depends on the Caputo fractional derivative for each step in obtaining the coefficient expansion. To test the simplicity, performance, and applicability of the present method, three numerical problems of the time-fractional Swift-Hohenberg initial value problems are considered. The impact of the fractional order β on the behavior of the approximate solutions at fixed bifurcation parameter is shown graphically and numerically. Obtained results emphasized that the LRPS technique is an easy, efficient, and speed approach for the exact description of the linear and nonlinear time-fractional models that arise in natural sciences.

An efficient computational technique for local fractional Fokker Planck equation

The key aim of the present study is to compute the solution of local fractional Fokker Planck equation (LFFPE) on the Cantor set. We perform a comparison between the reduced differential transform method (RDTM) and local fractional series expansion method (LFSEM) employed to the LFFPE. The operators are considered in the local nature. The outcomes demonstrate the important characteristic of the two techniques which are very successful and simple to solve the differential equations having fractional derivative operator of local nature.

The local fractional series expansion solution for local fractional Korteweg-de Vries equation

In this paper, the local fractional series expansion method is used to find the series solution for the local fractional Korteweg-de Vries equation.

Initial-Boundary Value Problems for Local Fractional Laplace Equation Arising in Fractal Electrostatics

The initial-boundary value problems for the local fractional Laplace equation, which arises in fractal electrostatics, are investigated in this article. The non-differentiable solutions with different initial and boundary conditions are obtained by using the local fractional series expansion method.

Fractional series expansion method for fractional differential equations

Purpose– The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs).Design/methodology/approach– This method is based on the idea of Kantorovich method, convergent series, and the modified Riemann-Liouville derivative.Findings– This work suggests a new analytical technique. The FDEs are described in Jumarie’s sense.Originality/value– It finds a new method for solving linear FDEs. The solution procedure is elucidated by two examples.

Application of the Local fractional Adomian Decomposition and Series Expansion Methods for Solving Telegraph Equation on Cantor Sets

In this paper, we will compare between the local fractional Adomian decompositionmethod (LFADM) and local fractional series expansion method (LFSEM) for solving thetelegraph equation on Cantor sets within local fractional operator. The results obtainedby (LFADM) are compared with the results obtained by (LFSEM). Some examples aregiven to illustrate the efficiency and accuracy of the presented methods.

On the non‐newtonian viscous behavior of polymer melts in terms of temperature and pressure‐dependent hole fraction

ABSTRACTA new theoretical non‐Newtonian viscosity model is developed by taking the fractional series expansion of Eyring's shearing strain rate. A broad range of experimental rheological data of various polymer melts including polyethylenes, polypropylene, polystyrene, poly (methyl methacrylate), and polycarbonate are fitted well using the proposed model. From the model; zero shear, constant shear‐stress and constant shear‐rate viscosities are derived as a linear function of viscosity related quantity, Yh, called “thermo‐occupancy function” and their coefficients are discussed in detail. The thermo‐occupancy function is expressed in terms of temperature and structural vacancies such as hole fraction computed from the Simha‐Somcynsky Hole Theory (SS). In addition, the derivative of the logarithm of viscosities with respect to the hole fraction, named as viscoholibility, is observed decreases with the increasing hole fraction. © 2014 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2014, 131, 40540.

Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets

We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.

Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads

The combined effects of conservative and non-conservative loads on the mechanical behavior of an unshearable and inextensional visco-elastic beam, close to bifurcation, are investigated. The equations of motion and boundary conditions are derived via a constrained variational principle, and the Lagrange multiplier successively condensed, to get integro-differential equations. These latter, with the mechanical boundary conditions appended, are put in an operator-form, amenable to perturbation analysis. A linear stability analysis is carried out in the space of the two loading parameter, displaying the existence of codimension-1 and codimension-2 bifurcations. The influence of both internal and external damping on this scenario is thoroughly investigated. A post-critical analysis is carried out around a double-zero bifurcation, by using an adapted version of the multiple scale method, based on fractional series expansions in the perturbation parameter. The integro-differential problem is directly attacked, so that any a priori discretization is avoided. Emphasis is given to the interaction between the two damping coefficients. This reveals the existence, also in the non-linear range, of a phenomenon of destabilization, so far known only in the linear range.

Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators

We investigate solutions of the Helmholtz equation involving local fractional derivative operators. We make use of the series expansion method and the variational iteration method, which are based upon the local fractional derivative operators. The nondifferentiable solution of the problem is obtained by using these methods.

Approximation Solutions for Local Fractional Schrödinger Equation in the One-Dimensional Cantorian System

The local fractional Schrödinger equations in the one-dimensional Cantorian system are investigated. The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show that the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative.

Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

BIFURCATION ANALYSIS OF DAMPED VISCO-ELASTIC PLANAR BEAMS UNDER SIMULTANEOUS GRAVITATIONAL AND FOLLOWER FORCES

The mechanical behavior of a non-conservative non-linear beam, internally and externally damped, undergoing codimension-1 (static or dynamic) and codimension-2 (double-zero) bifurcations, is analyzed. The system consists of a purely flexible, planar, visco-elastic beam, fixed at one end, loaded at the tip by a follower force and a dead load, acting simultaneously. An integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the space of the two loading parameters. Attention is then focused on the double-zero bifurcation, for which a post-critical analysis is carried out without any a-priori discretization. An adapted version of the Multiple Scale Method, based on a fractional series expansion in the perturbation parameter, is employed to derive the bifurcation equations. Finally, bifurcation charts are evaluated, able to illustrate the system behavior around the codimension-2 bifurcation point.