Nonlinear Fractional Problem Research Articles

Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel

In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

Energy-Efficient Power Allocation for Distributed Antenna Systems With Proportional Fairness

In this paper, we propose an energy-efficient power allocation scheme for the downlink multiuser distributed antenna systems. The objective is to maximize the energy efficiency (EE) under the constraints on per-antenna transmit power and proportional data rates among users. Since EE function is typically defined in fractional form, it is computationally complex to optimize the EE directly. We first convert the nonlinear fractional problem into an equivalent but better tractable problem, based on which we derive an iterative algorithm to obtain the global optimum of the considered problem. On top of being optimal, the solution is also lightweight since only a single-variable nonlinear equation needs to be solved. Furthermore, we can flexibly switch to another mode of operation which delivers relatively higher spectral efficiency (SE) at the expense of losing EE. Numerical simulations and complexity analysis validate that the proposed scheme achieves much higher SE and EE performance with stricter satisfaction of the proportional rate constraints and lower complexity, compared to the state-of-the-art scheme in literature.

Existence results for nonlinear fractional Dirichlet problems on the right side of the first eigenvalue

Abstract In this paper, we are interested in the study of the existence of solutions of a class of nonlinear fractional differential equations coupled with Dirichlet boundary conditions. We make an exhaustive study of the sign of Green’s function related to the linear problem and obtain the exact values for which it has the constant sign on its whole square of definition. This constant sign is equivalent to the validity of anti-maximum principles. The existence of solutions is deduced from a recent fixed point theorem valid for operators defined on suitable cones of Banach spaces. Moreover, the method of lower and upper solutions is developed for the non-homogeneous Dirichlet problem.

Existence of three solutions for impulsive nonlinear fractional boundary value problems

In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results.

Unconditionally Convergent $L1$-Galerkin FEMs for Nonlinear Time-Fractional Schrödinger Equations

In this paper, a linearized $L1$-Galerkin finite element method is proposed to solve the multidimensional nonlinear time-fractional Schrodinger equation. In terms of a temporal-spatial error splitting argument, we prove that the finite element approximations in the $L^2$-norm and $L^\infty$-norm are bounded without any time-step size conditions. More importantly, by using a discrete fractional Gronwall-type inequality, optimal error estimates of the numerical schemes are obtained unconditionally, while the classical analysis for multidimensional nonlinear fractional problems always required certain time-step restrictions dependent on the spatial mesh size. Numerical examples are given to illustrate our theoretical results.

Unilateral Global Bifurcation, Half-Linear Eigenvalues and Constant Sign Solutions for a Fractional Laplace Problem

In this paper, we are concerned with the unilateral global bifurcation structure of fractional differential equation [Formula: see text] with nondifferentiable nonlinearity [Formula: see text]. It shows that there are two distinct unbounded subcontinua [Formula: see text] and [Formula: see text] consisting of the continuum [Formula: see text] emanating from [Formula: see text], and two unbounded subcontinua [Formula: see text] and [Formula: see text] consisting of the continuum [Formula: see text] emanating from [Formula: see text]. As an application of this unilateral global bifurcation results, we present the existence of the principal half-eigenvalues of the half-linear fractional eigenvalue problem. Finally, we deal with the existence of constant sign solutions for a class of fractional nonlinear problems. Main results of this paper generalize the known results on classical Laplace operators to fractional Laplace operators.

Energy efficiency based joint cell selection and power allocation scheme for HetNets

Heterogeneous networks (HetNets) composed of overlapped cells with different sizes are expected to improve the transmission performance of data service significantly. User equipments (UEs) in the overlapped area of multiple cells might be able to access various base stations (BSs) of the cells, resulting in various transmission performances due to cell heterogeneity. Hence, designing optimal cell selection scheme is of particular importance for it may affect user quality of service (QoS) and network performance significantly. In this paper, we jointly consider cell selection and transmit power allocation problem in a HetNet consisting of multiple cells. For a single UE case, we formulate the energy efficiency of the UE, and propose an energy efficient optimization scheme which selects the optimal cell corresponding to the maximum energy efficiency of the UE. The problem is then extended to multiple UEs case. To achieve joint performance optimization of all the UEs, we formulate an optimization problem with the objective of maximizing the sum energy efficiency of UEs subject to QoS and power constraints. The formulated nonlinear fractional optimization problem is equivalently transformed into two subproblems, i.e., power allocation subproblem of each UE-cell pair, and cell selection subproblem of UEs. The two subproblems are solved respectively through applying Lagrange dual method and Kuhn–Munkres (K-M) algorithm. Numerical results demonstrate the efficiency of the proposed algorithm.

Energy efficiency-based joint spectrum handoff and resource allocation algorithm for heterogeneous CRNs

Cognitive radio networks (CRNs) are expected to improve spectrum utilization significantly by allowing secondary users (SUs) to opportunistically access the licensed spectrum of primary users (PUs). In an integrated network consisting of multiple heterogeneous CRNs, SUs with multiple interfaces may have to conduct inter-system or intra-system spectrum handoff due to the arrival of PUs or performance degradation on serving spectrum. In this case, designing an optimal spectrum handoff scheme which offers quality of service (QoS) guarantee and performance enhancement of the SUs is of particular importance. On the other hand, resource allocation strategy on target channel also plays an important role in affecting the transmission performance of handoff SUs. In this paper, we jointly design spectrum handoff and resource allocation strategy for handoff SUs in heterogeneously integrated CRNs. To achieve joint resource management among various CRNs, we propose a joint radio resource management architecture, based on which the proposed spectrum handoff and resource allocation scheme can be conducted. Jointly considering the transmission performance of the handoff SUs, we formulate the total energy efficiency of the SUs and design an optimization problem which maximizes the energy efficiency subject to spectrum handoff, QoS, and power constraints of the SUs. An iterative algorithm is proposed to solve the formulated nonlinear fractional optimization problem. Within each iteration, the optimization problem is transformed equivalently into two subproblems, i.e., power allocation subproblem of each SU-spectrum pair and spectrum handoff subproblem for all the SUs. The two subproblems are solved, respectively, through applying Lagrange dual method and the Kuhn-Munkres (K-M) algorithm. Numerical results demonstrate the efficiency of the proposed algorithm.

Wireless Information and Power Transfer in Two-Way OFDM Amplify-and-Forward Relay Networks

Energy harvesting (EH) is promising to solve energy scarcity problem in energy-constrained wireless networks. In this letter, we investigate simultaneous power and information transfer in two-way amplify-and-forward orthogonal-frequency-division-multiplexing relay networks, where an energy-constrained relay node equipped EH device harvests energy from two terminals and uses the energy to forward information in a time-switching (TS) manner. By jointly designing TS ratios of EH and information processing at the relay, as well as power allocation over all the subcarriers at two terminals and relay, our objective is to maximize end-to-end achievable rate of two-way relay networks subject to transmit power constraint at two terminals and EH constraint at relay. The formulated problem is hard to tackle because the EH constraint is non-convex and the objective rate function has a highly non-convex structure. We propose to solve this non-convex problem by transforming it into a nonlinear fractional problem, which can be solved by successive convex approximation and Dinkelbach’s procedure, and thus propose an efficient iterative algorithm that achieves superior performance with fast convergence speed.

Positive solutions of higher-order nonlinear multi-point fractional equations with integral boundary conditions

Abstract In this article, we consider a nonlinear higher-order m-point fractional boundary-value problem with integral boundary conditions. We establish criteria for the existence of at least one, two and three positive solutions for higher order nonlinear m-point fractional boundary-value problems with integral boundary conditions by using the four functionals fixed point theorem, the Avery-Henderson fixed point theorem and the Legget-Williams fixed point theorem, respectively.

Numerical Computation of a Fractional Model of Differential-Difference Equation

In the present article, we apply a numerical scheme, namely, homotopy analysis Sumudu transform algorithm, to derive the analytical and numerical solutions of a nonlinear fractional differential-difference problem occurring in nanohydrodynamics, heat conduction in nanoscale, and electronic current that flows through carbon nanotubes. The homotopy analysis Sumudu transform method (HASTM) is an inventive coupling of Sumudu transform algorithm and homotopy analysis technique that makes the calculation very easy. The fractional model is also handled with the aid of Adomian decomposition method (ADM). The numerical results derived with the help of HASTM and ADM are approximately same, so this scheme may be considered an alternative and well-organized technique for attaining analytical and numerical solutions of fractional model of discontinued problems. The analytical and numerical results derived by the application of the proposed technique reveal that the scheme is very effective, accurate, flexible, easy to apply, and computationally very appropriate for such type of fractional problems arising in physics, chemistry, biology, engineering, finance, etc.

A computational approach for fractional convection-diffusion equation via integral transforms

Abstract In this paper, two efficient analytic techniques namely the homotopy analysis transform method (HATM) and homotopy perturbation Sumudu transform method (HPSTM) are implemented to give a series solution of fractional convection-diffusion equation which describes the flow of heat. These proposed techniques introduce significance in the field over the existing techniques that make them computationally very attractive for applications. Numerical solutions clearly demonstrate the reliability and efficiency of HATM and HPSTM to solve strongly nonlinear fractional problems.

A Full-Duplex Bob in the MIMO Gaussian Wiretap Channel: Scheme and Performance

This letter considers secrecy communication from an information-theoretic perspective, and studies the secrecy capacity of a multi-input multi-output (MIMO) Gaussian wiretap channel with a source (Alice), an eavesdropper (Eve) and a Full-Duplex (FD) legitimate receiver (Bob). Bob can allocate part of his antennas to transmit jamming signals to impair Eve’s channel. Our goal is to identify the secrecy capacity behavior in the high signal-to-noise ratio (SNR) regimes, i.e., the maximal achievable secure degrees of freedom (S.D.o.F). Such S.D.o.F maximization is generally difficult to solve since it requires to face a nonlinear fractional problem. To deal with this issue, we first propose a cooperative secrecy transmission scheme, and prove its optimality in the sense of achieving the maximal S.D.o.F.. By studying this proposed transmission scheme, we obtain the maximal achievable S.D.o.F. in closed form for any given antenna allocation at Bob. Based on this closed-form result, we further analytically derive the optimal antenna allocation at Bob. To the best of our knowledge, this is the first time that the benefit brought by using the FD jamming Bob has been quantified.

Residual power series method for fractional Sharma-Tasso-Olever equation

In this paper, we introduce a modified analytical approximate technique to obtain solution of time fractional Sharma-Tasso-Olever equation. First, we present an alternative framework of the Residual power series method (RPSM) which can be used simply and effectively to handle nonlinear fractional differential equations arising in several physical phenomena. This method is basically based on the generalized Taylor series formula and residual error function. A good result is found between our solution and the given solution. It is shown that the proposed method is reliable, efficient and easy to implement on all kinds of fractional nonlinear problems arising in science and technology.

Eigenvalue, Unilateral Global Bifurcation and Constant Sign Solution for a Fractional Laplace Problem

In this paper, we apply the Ljusternik–Schnirelmann theory to deal with the eigenvalues and eigenfunctions of the fractional Laplace operator. It shows that there exists a simple and isolated principal eigenvalue [Formula: see text] such that under certain conditions on the perturbation function [Formula: see text], [Formula: see text] is a bifurcation point of the problem [Formula: see text] where [Formula: see text], [Formula: see text] is a smooth and bounded domain, [Formula: see text] [Formula: see text] with [Formula: see text] and [Formula: see text] satisfies the Carathéodory condition in the first two variables. There are two distinct unbounded subcontinua [Formula: see text] and [Formula: see text], consisting of the continuum [Formula: see text] emanating from [Formula: see text]. As an application of the unilateral global bifurcation result, we extend our investigation to the existence of constant sign solutions for a class of related nonlinear fractional Laplace problems. Some known results on the fractional Laplace problem in the literature are generalized.

Differential Transform Method with Complex Transforms to Some Nonlinear Fractional Problems in Mathematical Physics

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.

Optimal resource allocation for energy efficiency in coordinated multicell OFDMA networks

SummaryBecause energy efficiency (EE) is inevitable in future wireless cellular networks, in this paper, we focus on improving the number of bits delivered to users for each unit energy consumption in the downlink of orthogonal frequency‐division multiple access cellular networks with base stations (BSs) coordination. Specifically, each BS shares the channel qualities of users with others and jointly choose the set of co‐channel users and the transmit power allocated to maximize the EE of the system subject to the transmit power ceiling of each BS and minimum required data rate. We formulate the problem as a nonlinear fractional optimization problem, using nonlinear fractional programming, the original hard‐to‐solve problem is transferred to a new one that has the same optimal solution and is easier to solve, this enables two iterative algorithms that achieve nearly the same maximum EE. Numerical results are provided to show the convergence and superiority of the two proposed. Copyright © 2014 John Wiley & Sons, Ltd.

Decode and forward relaying for energy‐efficient multiuser cooperative cognitive radio network with outage constraints

We investigate the optimal allocation of power in the downlink cooperative cognitive radio network using decode and forward (DF) relaying technique. The power allocation in DF relaying for green cooperative cognitive radio with an objective of maximising energy-efficiency is a constraint non-linear non-convex fractional programming problem. The optimisation needs to satisfy the primary users interference constraints and secondary users outage constraints. The authors present the optimal power allocation in DF relaying by transforming the constraint non-linear non-convex fractional power allocation problem into a concave fractional programme by using Charnes–Cooper transformation. The authors also present an iterative algorithm that uses parametric transformation and guarantees e-optimal convergence. The convergence of the iterative algorithm is proved and numerical results obtained for cooperative cognitive radio network are presented with different network parameter settings.

Application of Bessel functions for solving differential and integro-differential equations of the fractional order

In this paper, a new numerical algorithm to solve the linear and nonlinear fractional differential equations (FDE) is introduced. Fractional calculus and fractional differential equations have many applications in physics, chemistry, engineering, finance, and other sciences. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, which is convergent for any x∈R. In this method, we reduce the solution of a nonlinear fractional problem to the solution of a system of the nonlinear algebraic equations. To illustrate the reliability of this method, we solve some important equations of fractional order, and present numerical results of the present method to show convergence rate, applicability and reliability of this method.

Existence and Global Asymptotic Behavior of Positive Solutions for Nonlinear Fractional Dirichlet Problems on the Half-Line

We are interested in the following fractional boundary value problem:Dαu(t)+atuσ=0,t∈(0,∞),limt→0⁡t2-αu(t)=0,limt→∞⁡t1-αu(t)=0, where1<α<2,σ∈(-1,1),Dαis the standard Riemann-Liouville fractional derivative, andais a nonnegative continuous function on(0,∞)satisfying some appropriate assumptions related to Karamata regular variation theory. Using the Schauder fixed point theorem, we prove the existence and the uniqueness of a positive solution. We also give a global behavior of such solution.