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A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme. [ABSTRACT FROM AUTHOR]

Geiger-mode avalanche photodiode (GM-APD) light detection and ranging (LiDAR) target echo signals are susceptible to interference from atmospheric backscatter, daylight, and other noise, and extracted target depth images contain a large amount of anomalous noise. Accordingly, this paper devises a method based on a spatially correlated fractional integral for denoising GM-APD LiDAR depth images. First, a multi-scale superpixel fusion algorithm is employed to perform null-pixel complementation on a target depth image extracted using histogram statistics. Next, a pixel neighborhood spatial correlation kernel function is used to optimize the Grünwald–Letnikov fractional integral operator to correct the large amount of anomalous noise and thus achieve GM-APD depth image denoising. Simulation and experimental results demonstrate that the denoising performance of the algorithm on GM-APD LiDAR depth images is significantly better than that of median filtering and bilateral filtering, with at least 22.8 % and 4.8 % improvements in the target reduction degree (K) and peak signal-to-noise ratio (PSNR), respectively. In addition, the K and PSNR reach 91.36 % and 18.0241 dB, respectively, with 25 statistical frames in an outdoor imaging experiment. This demonstrates that the algorithm can effectively denoise GM-APD LiDAR depth images with few statistical frames and thus improve the frame rate of GM-APD LiDAR imaging. [ABSTRACT FROM AUTHOR]

In this paper, we present a continuous and discrete time weak Galerkin finite element method (WG-FEM) for solving two dimensional time fractional coupled Burgers' equations. The stability is proved for discrete time WG-FEM and the optimal order error in L 2 -norm is obtained based on fractional derivative definition, fractional integral definition and dual argument technique for continues and discrete time WG-FEM. The numerical example is to illustrate the theoretical analysis with polynomial mixture { P k (K) , P k − 1 (∂ K) , P k − 1 (K) 2 }. [ABSTRACT FROM AUTHOR]

We prove the boundedness of the fractional integration operator of variable order α (x) in the limiting Sobolev case α (x) p (x) = n − λ (x) from variable exponent Morrey spaces L p ⋅ , λ ⋅ Ω into BMO (Ω), where Ω is a bounded open set. In the case α (x) ≡ const, we also show the boundedness from variable exponent vanishing Morrey spaces V L p ⋅ , λ ⋅ Ω into VMO (Ω). The results seem to be new even when p and λ are constant. [ABSTRACT FROM AUTHOR]

Predicting the horizontal groundwater flow in unsaturated porous media is a challenge in many areas of science and engineering. The governing equation associated with this phenomenon is a nonlinear partial differential equation known as the Richards equation. However, the numerical results obtained using this equation can differ substantially from the experimental results. In order to overcome this difficulty, a new version of the Richards equation was proposed recently that considers a time derivative of fractional order. In this study, we present a numerical method for solving this fractional Richards equation. Our method comprises an adaptive time marching scheme that uses Picard iterations to solve the corresponding nonlinear equations. A computational code was implemented for the proposed method using the Scilab programming language. We performed numerical simulations of the anomalous diffusion of water in a white siliceous brick and showed that the numerical results were consistent with the available experimental data. [ABSTRACT FROM AUTHOR]

Let X be a ball quasi-Banach function space on R n satisfying some mild assumptions and let α ∈ (0 , n) and β ∈ (1 , ∞). In this article, when α ∈ (0 , 1) , the authors first find a reasonable version I ˜ α of the fractional integral I α on the ball Campanato-type function space L X , q , s , d (R n) with q ∈ [ 1 , ∞) , s ∈ Z + n , and d ∈ (0 , ∞). Then the authors prove that I ˜ α is bounded from L X β , q , s , d (R n) to L X , q , s , d (R n) if and only if there exists a positive constant C such that, for any ball B ⊂ R n , | B | α n ≤ C ‖ 1 B ‖ X β − 1 β , where X β denotes the β -convexification of X. Furthermore, the authors extend the range α ∈ (0 , 1) in I ˜ α to the range α ∈ (0 , n) and also obtain the corresponding boundedness in this case. Moreover, I ˜ α is proved to be the adjoint operator of I α. All these results have a wide range of applications. Particularly, even when they are applied, respectively, to mixed-norm Lebesgue spaces, Morrey spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all the obtained results are new. The proofs of these results strongly depend on the dual theorem on L X , q , s , d (R n) and also on the special atomic decomposition of molecules of H X (R n) (the Hardy-type space associated with X) which proves the predual space of L X , q , s , d (R n). [ABSTRACT FROM AUTHOR]

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new refinements and similar extensions of Hermite–Hadamard inequality for fractional integrals and present several applications in the Beta function. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a systematic study of some extended special functions like B b ; ρ , λ α , β x , y , 2 F 1 α , β ; ρ , λ z ; b and p F q α , β ; ρ , λ z ; b . We obtain various properties of these extended functions and establish their some connections with the Laguerre polynomial and Fox’s H -function. Furthermore, we also establish the extended Riemann–Liouville type fractional integral operator and extended Kober type fractional integral operators. [ABSTRACT FROM AUTHOR]

Abstract: We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for coincides with the classical definitions on polynomials (up to a constant). Further, if , the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, by using the Schauder fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and obtain some new results. [Copyright &y& Elsevier]

Abstract: Let be a Schrödinger operator with a non-negative potential V satisfying some appropriate reverse Hölder inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to L such as the Riesz transforms and fractional integrals with the new BMO functions introduced in Bongioanni et al. (2011) [1] on the weighted spaces where w belongs to the new classes of weights introduced by Bongioanni et al. (2011) [2]. [Copyright &y& Elsevier]

Abstract: Let , be a strongly Lipschitz domain in and a magnetic Schrödinger operator on satisfying the Dirichlet boundary condition, where and . In this paper, the authors introduce the Hardy space by the Lusin area function associated with and establish its equivalent characterization via the non-tangential maximal function associated with . As applications, the authors obtain the boundedness of the Riesz transforms , , from to for and the fractional integral from to for and , where is the closure of in . [Copyright &y& Elsevier]

Abstract: The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integro-differentiation. The fractional Maxwell equations are obtained, and methods of fractional calculus are employed to obtain analytical expressions of the electric field inside the fractal composite structures. [Copyright &y& Elsevier]

Abstract: In this paper, we first present a version of Filippov’s Theorem for fractional semilinear differential inclusions of the form, where is the Caputo fractional derivative and is a set-valued map. After several existence results, the compactness of solutions sets is also investigated. Some results from topological fixed point theory together with notions of measure of noncompactness are used. [Copyright &y& Elsevier]

Abstract: It is shown that, if the problem is defined in the setting of fractional calculus via fractional difference on non-differentiable functions, then the solution of the functional equation f(x + y)= f(x)f(y) is exactly defined as the solution of a linear fractional differential equation. The dual or counterpart problem, that is the fractional solution of the equation g(xy)= g(x)+ g(y), is also considered, and it is shown that the corresponding solution is the logarithm of fractional order defined as the inverse of a generalized Mittag–Leffler function which is nowhere differentiable. This framework suggests a definition of fractional Laplace’s transform expressed in terms of generalized Mittag–Leffler function, and its main properties are outlined: mainly inverse function and convolution. One takes this opportunity to display a (new) fractional Taylor’s series for functions f(x,y) of two variables x and y. Many open problems are stated, which are directly related to the non-differentiability of the functions so involved, therefore the title “on the fractional solution…”. [Copyright &y& Elsevier]

Abstract: By using the coincidence degree theory due to Mawhin and constructing suitable operators, we study the existence of solutions for a coupled system of fractional differential equations at resonance. An example is given to illustrate our result. [Copyright &y& Elsevier]

Abstract: In this work, we present the notion of the fractional resolvent, which can be seen as a generalization of strongly continuous semigroups. We give some of its properties and apply the results to a fractional order abstract evolution equation. [Copyright &y& Elsevier]

Abstract: By using the coincidence degree theory due to Mawhin and constructing suitable operators, we study the existence of solutions to boundary value problems of fractional differential equations at resonance with . An example is given to illustrate our result. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, by using fixed point theory and a new method, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and neutral differential equations with infinite delay and obtain two new existence and uniqueness results. [Copyright &y& Elsevier]

Abstract: Of concern is the following Cauchy problem for fractional integro-differential equations with time delay and nonlocal initial condition where , , is a generator of a solution operator on a complex Banach space , the convolution integral in the equation is known as the Riemann–Liouville fractional integral, representing the delay property, is a function, and is an operator defined from into for some which constitutes a nonlocal condition. The local existence and uniqueness of mild solutions for the Cauchy problem, under various criteria, are proved. Moreover, we present an existence result of the global solution. Also an example is given to illustrate the applications of the abstract results. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point theory, we study the existence and uniqueness of initial value problems for nonlinear fractional differential equations and obtain a new result. [Copyright &y& Elsevier]

Abstract: In the light of the fixed point theorems, we analytically establish the conditions for the uniqueness of solutions as well as the existence of at least one solution in the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equation. Furthermore, we provide a representative example to illustrate a possible application of the established analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential equations with variable times involving the Caputo fractional derivative. [Copyright &y& Elsevier]

Abstract: In this paper, we first present an impulsive version of the Filippov–Ważewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form where denotes the Caputo fractional derivative, and is a set-valued map. The functions characterize the jump of the solutions at impulse points . Additional existence results are obtained under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on the nonlinear alternative of Leray–Schauder type, a Bressan–Colombo selection theorem, and Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, sets, acyclicity and contractibility, corresponding to Aronszajn–Browder–Gupta type results, are obtained. We also consider the impulsive fractional differential equations and where is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions, where is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the Riemann–Liouville derivative. Our approach is based on integrated semigroup theory, the Banach contraction principle, and the nonlinear alternative of Leray–Schauder type. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions for the existence of solutions for a class of boundary value problem for fractional differential equations involving the Caputo fractional derivative and nonlocal conditions. [Copyright &y& Elsevier]

Abstract: We study -asymptotically -periodic solutions of the semilinear fractional equation , considered in a Banach space , where is a linear operator of sectorial type . [Copyright &y& Elsevier]

Abstract: In this paper, we present fractional versions of the Filippov theorem and the Filippov–Wazewski theorem, as well as an existence result, compactness of the solution set and Hausdorff continuity of operator solutions for functional differential inclusions with fractional order, where is the standard Riemann–Liouville fractional derivative, and is a set-valued map. [Copyright &y& Elsevier]

Abstract: In this paper, we study fractional differential inclusions with Dirichlet boundary conditions. We prove the existence of a solution under both convexity and nonconvexity conditions on the multi-valued right-hand side. The proofs rely on nonlinear alternative Leray–Schauder type, Bressan–Colombo selection theorem and Covitz and Nadler’s fixed point theorem for multi-valued contractions. The compactness of the set solutions and relaxation results is also established. In the last section we consider the fractional boundary value problem with infinite delay. [Copyright &y& Elsevier]

Abstract: We study almost automorphic (mild) solutions of the semilinear fractional equation , considered in a Banach space , where is a linear operator of sectorial type . We prove the existence and uniqueness of an almost automorphic mild solution assuming that is almost automorphic in for each , satisfies some Lipschitz type conditions and takes values on . [Copyright &y& Elsevier]

Abstract: In this paper, we consider the n-term linear fractional-order differential equation with constant coefficients and obtain the solution of this kind of fractional differential equations by Adomian decomposition method. With the equivalent transmutation, we show that the solution by Adomian decomposition method is the same as the solution by the Green''s function. Finally, we illustrate our result with some examples. [Copyright &y& Elsevier]

Abstract: In this article, an application of He''s variational iteration method is proposed to approximate the solution of a nonlinear fractional differential equation with Riemann–Liouville''s fractional derivatives. Also, the results are compared with those obtained by Adomian''s decomposition method and truncated series method. The results reveal that the method is very effective and simple. [Copyright &y& Elsevier]

Abstract: An algorithm has been developed to convert the multi-order fractional differential equation:where m < α ⩽ m +1, 0< β 1 < β 2 <⋯< β n < α and denotes Caputo fractional derivative of order α into a system of fractional differential equations. Further Adomian decomposition method has been employed to solve the system of fractional differential equations. Some illustrative examples are presented. [Copyright &y& Elsevier]

Abstract: In this paper, a new generalized Taylor’s formula of the kindwhere a j ∈ R , x > a, 0< α ⩽1, is established. Such an expression is precisely the classical Taylor’s formula in case of α =1. In addition, detailed expressions for and a j (α), that involving Caputo fractional derivatives, are well constructed. Some applications including approximation of functions and solutions of fractional differential equations are also given. [Copyright &y& Elsevier]

Abstract: In this paper, we present a powerful modification of the rectangular decomposition method that is used to solve fractional diffusion-wave equations. The new modification extends the application of the rectangular decomposition method to solve a wide class of space–time fractional partial differential equations. The proposed modification gives an analytical solution in the form of a convergent series with easily computable components. Some numerical examples are given to illustrate the efficiency and reliability of the method. The new modification introduces a promising tool for many partial differential equations. [Copyright &y& Elsevier]

Abstract: Adomian decomposition method has been employed to obtain solutions of a system of nonlinear fractional differential equations:denotes Caputo fractional derivative. Some examples are solved as illustrations, using symbolic computation. [Copyright &y& Elsevier]

Abstract: In this paper, a new modification of the Adomian decomposition method is effectively implemented for solving fractional diffusion-wave equations that will facilitate the calculations, where the fractional derivative is based on Caputo definition. The proposed algorithm gives an analytical solution in the form of a convergent series with easily computable components. Numerical examples are given to show the application of the present algorithm. The new modification introduces a promising tool for many partial differential equations. [Copyright &y& Elsevier]

Abstract: In this paper we propose two algorithms for numerical fractional integration and Caputo fractional differentiation. We present a modification of trapezoidal rule that is used to approximate finite integrals, the new modification extends the application of the rule to approximate integrals of arbitrary order α >0. We then, using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order α >0, where the fractional derivative based on Caputo definition, for a given function by a weighted sum of function and its ordinary derivatives values at specified points. The study is conducted through illustrative examples and error analysis. [Copyright &y& Elsevier]

Abstract: The existence of solutions to an integral equation of fractional order with hysteresis has been established. A fixed point theorem due to Schauder is the main tool in carrying out our proof. [Copyright &y& Elsevier]

Abstract: The aim of the present analysis is to apply Adomian decomposition method for the solution of a nonlinear fractional differential equation. Finally, the solution obtained by the decomposition method has been numerically evaluated and presented in the form of tables and then compared with those obtained by truncated series method. A good agreement of the results is observed. [Copyright &y& Elsevier]

Abstract: By making use of certain familiar integral operators, the authors introduce and investigate several new subclasses of starlike, convex, close-to-convex, and quasi-convex functions. Among other results presented here, the authors establish a number of inclusion relationships associated with some of these integral operators. Some of the results established in this paper would provide extensions of those given in earlier works. [Copyright &y& Elsevier]

We define the fractional derivate of order 0<α<1, induced by the Ornstein Uhlenbeck operator L, associated with respect to the Gaussian measure. We obtain a characterization of the Gaussian potential Lαp(γd) spaces, for 1 in terms of this operator and we study the relation between potential Lkp(γ1) spaces and Sobolev Wkp(γ1) spaces for 1 and k∈N. Also, we obtain a version of Calderon''s reproduction formula for the Gaussian measure in terms of this fractional derivate. [Copyright &y& Elsevier]

This paper discusses the analytical solutions of fractional partial differential equations using Integral Transform method. The fractional derivatives are considered with reference to modified Riemann–Liouville derivatives. Fractional Fourier transform (FrFT) is applied to solve fractional heat diffusion, fractional wave, fractional telegraph and fractional kinetic equations. The method proposed here is effective enough to work on these equations efficiently. [ABSTRACT FROM AUTHOR]