Fractional variational problems depending on indefinite integrals

We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and natural boundary conditions, which provide a generalization of the previous results found in the literature. Isoperimetric problems, problems with holonomic constraints and depending on higher-order Caputo derivatives, as well as fractional Lagrange problems, are considered.     

A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type

We consider a regularization for a class of discontinuous differential equations arising in the study of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to so-called “breaking points”, where the solution derivative is again discontinuous. Consequently, the problem of continuing the solution in a right neighborhood of a breaking point is equivalent to a Cauchy problem for an ode with a discontinuous right-hand side (see e.g. Bellen et al., 2009 [4]). Therefore a classical solution may cease to exist.The regularization is based on the replacement of the vector-field with its time average over an interval of length ε>0. The regularized solution converges as ε→⁰⁺ to the classical Filippov solution (Filippov, 1964, 1988 [13] and [14]). Several properties of the solutions corresponding to small ε>0 are presented.     

Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.     

A fractional version of the peano-sard theorem

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.

In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.

The standard version of the Peano-Sard theory is contained in our results as a special case.     

Positivity of the Shape Hessian and Instability of some Equilibrium Shapes

We study the positivity of the second shape derivative around an equilibrium for a 2-dimensional functional involving the perimeter of the shape and its the Dirichlet energy under volume constraint. We prove that, generally, convex equilibria lead to strictly positive second derivatives. We also exhibit some examples where strict positivity of the second order derivative holds at an equilibrium while existence of a minimum does not.     

Cauchy problem for fractional diffusion equations

We consider an evolution equation with the regularized fractional derivative of an order α∈(0,1) with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated.     

Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions

In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann–Liouvile definition of fractional derivatives, one (Jumarie) has proposed recently an alternative referred to as a modified Riemann–Liouville definition, which directly, provides a Taylor’s series of fractional order for non differentiable functions. This fractional derivative provides a fractional calculus parallel with the classical one, which applies to non-differentiable functions; and the present short article summarizes the main basic formulae so obtained.     

Nonlinear dynamic analysis of viscoelastic beams using a fractional rheological model

The paper presents quasi-static analysis, classical and fractional dynamic analysis of a simply supported viscoelastic beam subjected to uniformly distributed load, where the Riemann–Liouville fractional derivative is of the order ν ∈ (0, 1). A comparative study of the results obtained for a classical and fractional Zener model using the techniques of Laplace transform, Bessel functions theory and binomial series is achieved. The graphic representations show how the existence of fractional derivative in the selected rheological model influences the dynamic response of the structure. This paper provides a theoretical basis for researchers who want to choose a mathematical model that will precisely fit with a particular experimental model.     

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis.     

Analysis of projectile motion in view of fractional calculus

The projectile motion is examined by means of the fractional calculus. The fractional differential equations of the projectile motion are introduced by generalizing Newton’s second law and Caputo’s fractional derivative is considered to use the physical initial conditions. In the absence of air resistance it is found that at certain conditions, the range and the maximum height of the projectile obtained by using the fractional calculus give the same results of the classical calculus. It is also found that, unlike the classical projectile motion, the launching angle that maximizes the horizontal range is a function of the arbitrary order of the fractional derivative α. Moreover, in a resistant medium, the parametric equations are expressed in terms of Mittag-Leffler function and the results agree with those of the classical projectile as α → 2. Moreover, the trajectories of the projectile are discussed in graphs and compared with those of the classical calculus. In order to explore the validity of modelling the projectile motion by the fractional approach, we compared our results with the experimental data of mortar.     

Necessary conditions of the isolated zero-points with nonzero degree and choice of initial approximations for the secant method in the n-dimensional case

Necessary conditions of isolated zero-points with nonzero degree (in particular, point of local minimum or maximum either saddlepoint) in R ⁿ which is an extension of the rule that the first derivative of a function changes a sign in every neighborhood of an isolated point of local minimum or maximum in JR1 and connected with this choice of initial approximations for the secant method are given and proved. Most of the statements are given through using the basic topological facts such as degree, the fiberings of a finite dimensional sphere, homotopy, quotient topology, etc.     

Mittag–Leffler Functions and the Truncated $${\mathcal {}}$$-fractional Derivative

In this paper, we introduce a new type of fractional derivative, which we called truncated \({\mathcal {V}}\)-fractional derivative, for \(\alpha \)-differentiable functions, by means of the six-parameter truncated Mittag–Leffler function. One remarkable characteristic of this new derivative is that it generalizes several different fractional derivatives, recently introduced: conformable fractional derivative, alternative fractional derivative, truncated alternative fractional derivative, M-fractional derivative and truncated M-fractional derivative. This new truncated \({\mathcal {V}}\)-fractional derivative satisfies several important properties of the classical derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag–Leffler function is a generalization of Mittag–Leffler functions of one, two, three, four and five parameters, we were able to extend some of the classical results of the integer-order calculus, namely: Rolle’s theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for derivatives and as an application we calculate the truncated \({\mathcal {V}}\)-fractional derivative of the two-parameter Mittag–Leffler function. Finally, we present the \({\mathcal {V}}\)-fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the \({\mathcal {V}}\)-fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated \({\mathcal {V}}\)-fractional derivative and the truncated \({\mathcal {V}}\)-fractional integral and the fractional derivative and fractional integral in the Riemann–Liouville sense when the order parameter \(\alpha \) lies between 0 and 1 (\(0<\alpha <1\)).     

Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

An inverse problem of determining a time‐dependent source term from the total energy measurement of the system (the over‐specified condition) for a space‐time fractional diffusion equation is considered. The space‐time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional‐order time derivative and Sturm‐Liouville operator by fractional‐order Sturm‐Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.     

Integral equations and initial value problems for nonlinear differential equations of fractional order

We discuss the solvability of integral equations associated with initial value problems for a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative and the inhomogeneous term depends on the fractional derivative of lower orders. We obtain the existence of at least one solution for integral equations using the Leray–Schauder Nonlinear Alternative for several types of initial value problems. In addition, using the Banach contraction principle, we establish sufficient conditions for unique solutions. Our approach in obtaining integral equations is the “reduction” of the fractional order of the integro-differential equations based on certain semigroup properties of the Caputo operator.     

Travelling waves for a non-local Korteweg–de Vries–Burgers equation

We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV–Burgers equation are not in general monotone, as is the case for the corresponding classical KdV–Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV–Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone.     

复杂介质中扩散和耗散行为的分数阶导数唯象建模

复杂介质一般是多相混合物.与普通固体、液体和气体相比,其力学行为具有明显的记忆、路径依赖性特征,难以用一般的经典力学模型来描述,因而显得反常.从数学力学建模上看,整数阶导数的局部极限定义不适合描述这样的非局部力学行为.分数阶导数实质上是微分-积分算子,能精确地刻画力学行为的全局相关特征.而且分数阶模型具有明确的统计物理解释.20世纪末至今,复杂介质反常力学行为的分数阶导数模型由于具有参数少,且参数的物理意义明确等突出优点,开始引起广泛关注.该文从唯象建模的角度,综述了分数阶导数和分形导数在复杂介质的反常扩散和频率依赖能量耗散建模中的应用与发展.     

Strict feasibility and stable solvability of bifunction variational inequalities

In this paper, we study strict feasibility of a bifunction variational inequality. It is proved that a monotone bifunction variational inequality has a nonempty and bounded solution set if and only if it is strictly feasible. Stable solvability of the bifunction variational inequality is discussed under strict feasibility assumption when the domain set is perturbed. Our results generalize earlier results on the classical variational inequality to the case of the bifunction variational inequality.     

Maximum principles for a time–space fractional diffusion equation

In this paper, we focus on maximum principles of a time–space fractional diffusion equation. Maximum principles for classical solution and weak solution are all obtained by using properties of the time fractional derivative operator and the fractional Laplace operator. We deduce maximum principles for a full fractional diffusion equation, other than time-fractional and spatial-integer order diffusion equations.     

The fundamental solution of the space-time fractional advection-dispersion equation

A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order α ∈ (0, 1], and the second-order space derivative is replaced with a Riesz-Feller derivative of order β ∈ (0, 2]. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.     

Stability of critical shapes for the drag minimization problem in Stokes flow

We study the stability of some critical (or equilibrium) shapes in the minimization problem of the energy dissipated by a fluid (i.e. the drag minimization problem) governed by the Stokes equations. We first compute the shape derivative up to the second order, then provide a sufficient condition for the shape Hessian of the energy functional to be coercive at a critical shape. Under this condition, the existence of such a local strict minimum is then proved using a precise upper bound for the variations of the second order shape derivative of the functional with respect to the coercivity and differentiability norms. Finally, for smooth domains, a lower bound of the variations of the drag is obtained in terms of the measure of the symmetric difference of domains.