Stability of the fractional Volterra integro‐differential equation by means of ψ‐Hilfer operator

In this paper, using the Riemann‐Liouville fractional integral with respect to another function and the ψ?Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro‐differential equation. In this sense, for this new fractional Volterra integro‐differential equation, we study the Ulam‐Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed‐point theorem. As an application, we present the Ulam‐Hyers stability using the α‐resolvent operator in the Sobolev space .     

Concentration phenomena for a fractional Schrödinger‐Kirchhoff type equation

In this paper, we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger‐Kirchhoff type equation where ε>0 is a small parameter, is the fractional Laplacian, M is a Kirchhoff function, V is a continuous positive potential, and f is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik‐Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum.     

Nonlocal boundary value problem for fractional differential equations with p‐Laplacian

In this paper, we are concerned with the existence of positive solutions for the following nonlocal BVP of fractional DEs with p‐Laplacian operator By using the fixed point theorem in a cone, multiplicity solutions of the BVP are obtained. An example is also given to show the effectiveness of the obtained result. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of solutions of periodic‐type boundary value problems for multi‐term fractional differential equations

Results on the existence of solutions of a periodic‐type boundary value problem of singular multi‐term fractional differential equations with the nonlinearity depending on are established and being singular at t = 0 and t = 1. The analysis relies on the well‐known fixed‐point theorems. An example is given to illustrate the efficiency of the main theorems. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions to space‐time fractional diffusion‐reaction equations

This article discusses the analyticity and the long‐time asymptotic behavior of solutions to space‐time fractional diffusion‐reaction equations in . By a Laplace transform argument, we prove that the decay rate of the solution as t→∞ is dominated by the order of the time‐fractional derivative. We consider the decay rate also in a bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three‐point fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form: where 0 < α,β≤1, 1 < α + β≤2, λ and ρ are constants, γ > 0, , is a continuous function, and E_βx(t) = x(t + β ? 1). The existence and uniqueness of solutions are proved by using Banach's fixed point theorem. An illustrative example is also presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Boundary value problem for p−Laplacian Caputo fractional difference equations with fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form where 0 < α,β≤1, 1 < α + β≤2, 0 < γ≤1, , ρ is a constant, and denote the Caputo fractional differences of order α and β, respectively, is a continuous function, and ?_p is the p‐Laplacian operator. The existence of at least one solution is proved by using Banach fixed point theorem and Schaefer's fixed point theorem. Some illustrative examples are also presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of nontrivial solutions for fractional Schrödinger‐Poisson system with sign‐changing potentials

In this paper, we consider the following fractional Schrödinger‐Poisson system where 0 < t ≤ α < 1, , and 4α+2t ≥3 and the functions V(x), K(x) and f(x) have finite limits as |x|→∞. By imposing some suitable conditions on the decay rate of the functions, we prove that the above system has two nontrivial solutions. One of them is positive and the other one is sign‐changing. Recent results from the literature are generally improved and extended.     

Exact solutions of some systems of fractional differential‐difference equations

In this paper, the ‐expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential‐difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential‐difference equation into its differential‐difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time‐fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd.     

One‐dimensional q‐Dirac equation

In this paper, we introduce a q‐analog of 1‐dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self‐adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of . Finally, we give some examples.     

Ground state of solutions for a class of fractional Schrödinger equations with critical Sobolev exponent and steep potential well

In this paper, we study the following fractional Schrödinger equations: (1) where (?△)^α is the fractional Laplacian operator with , 0≤s ≤2α , λ >0, κ and β are real parameter. is the critical Sobolev exponent. We prove a fractional Sobolev‐Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.     

Nonexistence results for the Cauchy problem of time fractional nonlinear systems of thermo‐elasticity

We study the nonexistence of global solutions to the Cauchy problem for systems of time fractional parabolic‐hyperbolic and time fractional hyperbolic thermo‐elasticity equations in . For certain nonlinearities, we present ‘threshold’ exponents depending on the space dimension d. Our proof rests on the test function method. Copyright © 2017 John Wiley & Sons, Ltd.     

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e_?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

Three rapidly convergent parareal solvers with application to time‐dependent PDEs with fractional Laplacian

Time‐dependent PDEs with fractional Laplacian ( ? Δ)^α play a fundamental role in many fields and approximating ( ? Δ)^α usually leads to ODEs' system like u ^′ (t ) + A u (t ) =  g (t ) with A  = Q ^α , where is a sparse symmetric positive definite matrix and α  > 0 denotes the fractional order. The parareal algorithm is an ideal solver for this kind of problems, which is iterative and is characterized by two propagators and . The propagators and are respectively associated with large step size ΔT and small step size Δt , where ΔT  = J Δt and J ?2 is an integer. If we fix the ‐propagator to the Implicit‐Euler method and choose for some proper Runge–Kutta (RK) methods, such as the second‐order and third‐order singly diagonally implicit RK methods, previous studies show that the convergence factors of the corresponding parareal solvers can satisfy and , where σ (A ) is the spectrum of the matrix A . In this paper, we show that by choosing these two RK methods as the ‐propagator, the convergence factors can reach , provided the one‐stage complex Rosenbrock method is used as the ‐propagator. If we choose for both and , the complex Rosenbrock method, we show that the convergence factor of the resulting parareal solver can also reach . Numerical results are given to support our theoretical conclusions. Copyright © 2017 John Wiley & Sons, Ltd.     

Reciprocity formulae for generalized Dedekind‐Vasyunin‐cotangent sums

Recently, several works are done on the generalized Dedekind‐Vasyunin sum where and q are positive coprime integers, and ζ(a,x) denotes the Hurwitz zeta function. We prove explicit formula for the symmetric sum which is a new reciprocity law for the sum . Our result is a complement to recent results dealing with the sum studied by Bettin‐Conrey and then by Auli‐Bayad‐Beck. Accidentally, when a = 0, our reciprocity formula improves the known result in a previous study.     

Analytical solution for a generalized space‐time fractional telegraph equation

In this paper, we consider a nonhomogeneous space‐time fractional telegraph equation defined in a bounded space domain, which is obtained from the standard telegraph equation by replacing the first‐order or second‐order time derivative by the Caputo fractional derivative , α > 0 and the Laplacian operator by the fractional Laplacian ( ? Δ)^{β ∕ 2}, β ∈ (0,2]. We discuss and derive the analytical solutions under nonhomogeneous Dirichlet and Neumann boundary conditions by using the method of separation of variables. The obtained solutions are expressed through multivariate Mittag‐Leffler type functions. Special cases of solutions are also discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of positive solutions to a coupled system of fractional differential equations

We consider the following system of fractional differential equations where is the Riemann‐Liouville fractional derivative of order α,f,g : [0,1] × [0, ∞ ) × [0, ∞ ) → [0, ∞ ). Sufficient conditions are provided for the existence of positive solutions to the considered problem. Copyright © 2014 John Wiley & Sons, Ltd.     

A mixed‐type Galerkin variational formulation and fast algorithms for variable‐coefficient fractional diffusion equations

We consider the variable‐coefficient fractional diffusion equations with two‐sided fractional derivative. By introducing an intermediate variable, we propose a mixed‐type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over . On the basis of the formulation, we develop a mixed‐type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz‐like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from O(N²) to O(N) and the computing cost from O(N³) to O(NlogN). Numerical experiments are included to verify our theoretical findings. Copyright © 2017 John Wiley & Sons, Ltd.     

On the concept of exact solution for nonlinear differential equations of fractional‐order

In this article, the new exact travelling wave solutions of the nonlinear space‐time fractional Burger's, the nonlinear space‐time fractional Telegraph and the nonlinear space‐time fractional Fisher equations have been found. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into ordinary differential equations of integer order in the sense of the Jumarie's modified Riemann–Liouville derivative. The ‐expansion method is effective for constructing solutions to the nonlinear fractional equations, and it appears to be easier and more convenient by means of a symbolic computation system. Copyright © 2016 John Wiley & Sons, Ltd.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.