Evaluating higher derivative tensors by forward propagation of univariate Taylor series

This article considers the problem of evaluating all pure and mixed partial derivatives of some vector function defined by an evaluation procedure. The natural approach to evaluating derivative tensors might appear to be their recursive calculation in the usual forward mode of computational differentiation. However, with the approach presented in this article, much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate Taylor series of a suitable degree. It is applicable for arbitrary orders of derivatives. Also it is possible to calculate derivatives only in some directions instead of the full derivative tensor. Explicit formulas for all tensor entries as well as estimates for the corresponding computational complexities are given.

     

A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel function. Several properties related to this new operator, like inversion rules and integration by parts, are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann‐Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.     

Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative

A Hamilton–Jacobi equation with Caputo’s time fractional derivative of order less than one is considered. The notion of a viscosity solution is introduced to prove unique existence of a solution to the initial value problem under periodic boundary conditions. For this purpose, comparison principle as well as Perron’s method is established. Stability with respect to the order of derivative as well as the standard one is studied. Regularity of a solution is also discussed. Our results in particular apply to a linear transport equation with time fractional derivatives with variable coe?cients.     

用Adomian分解法求解分数阻尼梁的解析解

利用Adomian分解法, 得到了由任意阶分数微分描述的具有阻尼特性的黏弹性连续梁的解析解．解中包含了任意的初始条件和零输入．为了更明确的分析, 假定初始条件是奇次的,输入受力是针对某种特定梁的特殊过程．分别考虑了两种简单情况下梁的响应:阶跃激励和脉冲激励．然后在系统的不同组参数条件下绘制了梁的位移图,并且讨论了梁在不同微分阶数下响应情况．     

On calculus of local fractional derivatives

Local fractional derivative (LFD) operators have been introduced in the recent literature (Chaos 6 (1996) 505-513). Being local in nature these derivatives have proven useful in studying fractional differentiability properties of highly irregular and nowhere differentiable functions. In the present paper we prove Leibniz rule, chain rule for LFD operators. Generalization of directional LFD and multivariable fractional Taylor series to higher orders have been presented.     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

Caputo–Hadamard Fractional Derivatives of Variable Order

In this article, we present three types of Caputo–Hadamard derivatives of variable fractional order and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is obtained and an estimation for the error is given. At the end, we compare the exact fractional derivative of a concrete example with some numerical approximations.     

Persymmetric determinants

The object of this paper is to develop some of the results in the author's joint paper with Dale [2] concerning the derivatives of persymmetric determinants whose elements are Appell functions.Four new double-sum identities are presented which are valid for arbitrary persymmetric determinants. Two of these identities are applied to give direct proofs of two results in [2], A simple formula is given for the derivative of a Turanian of order n with Appell polynomial elements and the result is applied repeatedly to show that its degree is far lower than expected. It is shown that one particular determinant has simple derivatives of all orders and that its degree too is far lower than expected. The formula for the derivative of (first) cofactors is shown to be extensible in a simple manner to the derivatives of second cofactors.     

Quenching of combustion explosion model with balanced space-fractional derivative

Balanced space-fractional derivative is usually applied in modelling the state-dependence, isotropy, and anisotropy in diffusion phenomena. In this paper, we introduce a class of space-fractional reaction-diffusion model with singular source term arising in combustion process. The fractional derivative employed in this model is defined in the sum of left-sided and right-sided Riemann-Liouville fractional derivatives. With assistance of Kaplan's first eigenvalue method, we prove that the classic solution of this model may not be globally well-defined, and the heat conduction governed by this model depends on the order of fractional derivative, the parameters in the equation, and the length of spatial interval. Finite difference method is implemented to solve this model, and an adaptive strategy is applied to improve the computational efficiency. The positivity, monotonicity, and stability of the numerical scheme are discussed. Numerical simulation and observation of the quenching and stationary solutions coincide the theoretical studies.     

Consistent initialization for DAEs in Hessenberg form

For nonlinear DAEs, we can hardly make a reasonable statement unless structural assumptions are given. Many results are restricted to explicit DAEs, often in Hessenberg form of order up to three. For the DAEs resulting from circuit simulation, different beneficial structures have been found and exploited for the computation of consistent initial values. In this paper, a class of DAEs in nonlinear Hessenberg form of arbitrary high order is defined and analyzed with regard to consistent initialization. For this class of DAEs, the hidden constraints can be systematically described and the consistent initialization can be determined step-by-step solving linear subproblems, an approach hitherto used for the DAEs resulting from circuit simulation. Finally, it is shown that the DAEs resulting from mechanical systems fulfill the defined structural assumptions. The algorithm is illustrated by several examples.     

Graph coloring in the estimation of sparse derivative matrices: Instances and applications

We describe a graph coloring problem associated with the determination of mathematical derivatives. The coloring instances are obtained as intersection graphs of row partitioned sparse derivative matrices. The size of the graph is dependent on the partition and can be varied between the number of columns and the number of nonzero entries. If solved exactly our proposal will yield a significant reduction in computational cost of the derivative matrices. The effectiveness of our approach is demonstrated via a practical problem from computational molecular biology. We also remark on the hardness of the generated coloring instances.     

High‐order compact schemes for fractional differential equations with mixed derivatives

In this article, we consider two‐dimensional fractional subdiffusion equations with mixed derivatives. A high‐order compact scheme is proposed to solve the problem. We establish a sufficient condition and show that the scheme converges with fourth order in space and second order in time under this condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2141–2158, 2017     

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

As the generalization of the integer order partial differential equations (PDE), the fractional order PDEs are drawing more and more attention for their applications in fluid flow, finance and other areas. This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin (DG) methods for one- and two-dimensional fractional diffusion equations containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative, because it may represent the fractional derivative by an integral operator. Some numerical examples show that the convergence orders of the proposed local $P^k$-DG methods are $O(h^{k+1})$ both in one and two dimensions, where $P^k$ denotes the space of the real-valued polynomials with degree at most $k$.     

A mathematical fractional model with nonsingular kernel for thrombin receptor activation in calcium signalling

In calcium signalling, activation of receptor is a very significant aspect. To understand the mechanism of calcium signalling, receptors are the important components. The mobilization of intracellular calcium from intracellular stores depends upon binding of agonist to cell surface receptor. Thrombin is chosen as model ligand. In order to understand thrombin receptor activation, we analyze fractional model incorporating derivative of arbitrary order and nonsingular kernel which can precisely describe the effect of memory and can explain the model in better and more efficient manner as compared with fractional operators with singular kernels. The problem has been solved by perturbation iterative method. Using fixed‐point theorem, it is proved that solution of the system will exist and also it will be unique.     

Higher-order optimality conditions for strict local minima

In this work, we study a nonsmooth optimization problem with generalized inequality constraints and an arbitrary set constraint. We present necessary conditions for a point to be a strict local minimizer of order k in terms of higher-order (upper and lower) Studniarski derivatives and the contingent cone to the constraint set. In the same line, when the initial space is finite dimensional, we develop sufficient optimality conditions. We also provide sufficient conditions for minimizers of order k using the lower Studniarski derivative of the Lagrangian function. Particular interest is put for minimizers of order two, using now a special second order derivative which leads to the Fréchet derivative in the differentiable case.     

On Peano and Riemann derivatives

For a given real function of one real variablef the conceptsn-th order Peano and (modified) Riemann derivatives are introduced. The conjecture is formulated that Peano derivative ofn-th order exists if and only if all Riemann derivatives of order less or equal ton exist and then then-th order Peano and Riemann derivative coincide. It is shown that this conjecture is equivalent to an assertion about the value of certain functional determinant of order 2n+1. This assertion is checked forn≤8. The general case remains an open question.     

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.