Higher order numerical methods for solving fractional differential equations

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0<α<1. The order of convergence of the numerical method is O(h ^3?α ). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α>0. The order of convergence of the numerical method is O(h ³) for α≥1 and O(h ^1+2α ) for 0<α≤1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.     

Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces

In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space E $$\left\{ \begin{gathered} u'(t) + Au(t) = f(t,u(t),Gu(t)) t \in J,t \ne t_k , \hfill \\ \Delta _{\left. u \right|_{t = t_k } } = u\left( {t_k^ + } \right) - u\left( {t_k^ - } \right) = I_k \left( {u\left( {t_k } \right)} \right), k = 1,2, \ldots ,m, \hfill \\ u(0) = g(u) + x_0 , \hfill \\ \end{gathered} \right.$$ where A: D(A) ? E → E is a closed linear operator and ?A generates a strongly continuous semigroup T(t) (t ? 0) on E, f ∈ C(J × E × E, E), J = [0, a], 0 < t ₁ < t ₂ < ... < t _m < a, I _k ∈ C(E, E), k = 1, 2, ..., m, and g constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that ?A generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered.     

Complete asymptotic analysis of positive solutions of odd-order nonlinear differential equation

We study the asymptotic behavior of solutions of the odd-order differential equation of Emden–Fowler type $$ {x^{{\left( {2n+1} \right)}}}(t)=q(t){{\left| {x(t)} \right|}^{\gamma }}\operatorname{sgn}x(t) $$ in the framework of regular variation under the assumptions that 0 < γ < 1 and q(t) : [a, ∞) → (0, ∞) is regularly varying function. We show that complete and accurate information can be acquired about the existence of all possible positive solutions and their asymptotic behavior at infinity.     

对一个非线性分数阶微分方程组爆破解的研究

The paper focuses on the blow-up solution of system of time-fractional differential equations
where ^cD₀₊^α, ^cD₀₊^β are Caputo fractional derivatives, n-1 < α < n, n-1 < β < n,A(t),B(t) are continuous functions. We obtain a system of the integral equations which is equivalent to the system of nonlinear partial differential equations with time-fractional derivative via the approach of Laplace transformation, and prove the local existence of solutions to the system of the integral equations. Secondly, this paper investigates the blow-up solutions to the a nonlinear system of fractional differential equations by making use of Hölder’s inequality and obtains a solution of system to blow up in a finite time, and gives an upper bound on the blow-up time.     

On FBDF5 Method for Delay Differential Equations of Fractional Order with Periodic and Anti-Periodic Conditions

In this paper, we study the fractional backward differential formula (FBDF) for the numerical solution of fractional delay differential equations (FDDEs) of the following form: \(\lambda _n {}_0^C D_t^{\alpha _n } y(t - \tau ) + \lambda _{n - 1} {}_0^C D_t^{\alpha _{n - 1} } y(t - \tau ) + \cdots + \lambda _1 {}_0^C D_t^{\alpha _1 } y(t - \tau ) + \lambda _{n + 1} y(t) = f(t), t \in [0,T]\), where \( \lambda _i \in \) \(\mathbb {R}\,(i = 1,\ldots ,n + 1)\,,\,\lambda _{n + 1} \ne 0,\,\, 0 \leqslant \alpha _1< \alpha _2< \cdots< \alpha _n < 1,\,\,T > 0,\) in Caputo sense. We find the Green’s functions for this equation corresponding to periodic/anti-periodic conditions in term of the Mittag-Leffler type. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the FDDEs. Finally, some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis     

Bounded mild solutions to fractional integro-differential equations in Banach spaces

We study the existence and uniqueness of bounded solutions for the semilinear fractional differential equation $$D^\alpha u(t)= Au(t)+ \int_{-\infty}^t a(t-s)Au(s)ds+ f \bigl(t,u(t) \bigr), \quad t \in\mathbb{R}, $$ where A is a closed linear operator defined on a Banach space X, α>0, a∈L ¹(?₊) is a scalar-valued kernel and f:?×X→X satisfies some Lipschitz type conditions. Sufficient conditions are established for the existence and uniqueness of an almost periodic, almost automorphic and asymptotically almost periodic solution, among other.     

Multiple positive solutions for fractional differential systems

In this paper, we study the existence of positive solution to boundary value problem for fractional differential system $$\left\{\begin{array}{ll}D_{0^+}^\alpha u (t) + a_1 (t) f_1 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1),\\D_{0^+}^\alpha v (t) + a_2 (t) f_2 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1), \;\; 2 < \alpha < 3,\\u (0)= u' (0) = 0, \;\;\;\; u' (1) - \mu_1 u' (\eta_1) = 0,\\v (0)= v' (0) = 0, \;\;\;\; v' (1) - \mu_2 v' (\eta_2) = 0,\end{array}\right.$$ where ${D_{0^+}^\alpha}$ is the Riemann-Liouville fractional derivative of order ??. By using the Leggett-Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.     

Solvability of boundary-value problems for nonlinear fractional differential equations

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations

On the stochastic response of a fractionally-damped Duffing oscillator

A numerical method is presented to compute the response of a viscoelastic Duffing oscillator with fractional derivative damping, subjected to a stochastic input. The key idea involves an appropriate discretization of the fractional derivative, based on a preliminary change of variable, that allows to approximate the original system by an equivalent system with additional degrees of freedom, the number of which depends on the discretization of the fractional derivative. Unlike the original system that, due to the presence of the fractional derivative, is governed by non-ordinary differential equations, the equivalent system is governed by ordinary differential equations that can be readily handled by standard integration methods such as the Runge–Kutta method. In this manner, a significant reduction of computational effort is achieved with respect to the classical solution methods, where the fractional derivative is reverted to a Grunwald–Letnikov series expansion and numerical integration methods are applied in incremental form. The method applies for fractional damping of arbitrary order α (0 < α < 1) and yields very satisfactory results. With respect to its applications, it is worth remarking that the method may be considered for evaluating the dynamic response of a structural system under stochastic excitations such as earthquake and wind, or of a motorcycle equipped with viscoelastic devices on a stochastic road ground profile.     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,     

On splitting the area under curves

Two area-splitting problems involving real-valued functions of a real variable are investigated. The second of these is essentially equivalent to finding all functions a?C¹((0, r)) with 0 < a(x) < x which satisfy the functional differential equation $\text{a′ (a(x)) =}\text{a(x)}\text{x}\text{for}\text{x ? (0, r)}$. All solutions analytic at x = 0 (and many which are not) are exhibited in closed form.     

Oscillation of third-order nonlinear functional differential equations with mixed arguments

The objective of this paper is to offer sufficient conditions for the oscillation of all solutions and other asymptotic properties of the third-order nonlinear functional differential equation $$\left[a(t)\left[x''(t)\right]^{\gamma}\right]' =q(t)f \left(x\left[\tau(t)\right]\right)+p(t)h \left(x\left[\sigma(t)\right]\right)$$ with mixed arguments, where both cases ∫^∞ a ^?1/γ (s)?ds=∞ and ∫^∞ a ^?1/γ (s)?ds<∞ are dealt with. We deduce properties of the studied equations via new comparison theorems. Our results essentially improve and complement earlier ones. We also repair one interesting result of Grace et al.     

Mild and strong solutions to few types of fractional order nonlinear equations with periodic boundary conditions

In this paper, the two fractional periodic boundary value problems $$_0^C D_{0 + }^\alpha u\left( t \right) - \lambda u\left( t \right) = f\left( {t,u\left( t \right)} \right), u\left( 0 \right) = u\left( 1 \right), 0 < \alpha < 1,$$ and $$_0^C D_{0 + }^\beta u\left( t \right) - \lambda u\left( t \right) = f\left( {t,u\left( t \right)} \right), u\left( 0 \right) = u\left( 1 \right),u'\left( 0 \right) = 0 1 < \beta < 2,$$ will be studied where ₀ ^C D _t ^α is the ordinary Caputo fractional derivative and λ ∈ ? ?{0}. Under some suitable assumptions on the function f, the existence of at least one mild solution will be proved. Under some conditions, the uniqueness of this mild solution will be proved to both problems. Finally, these mild solutions will be strong solutions under certain conditions.     

An abstract nonlinear Volterra integrodifferential equation

We study the nonlinear Volterra equation $\text{u′(t) + Bu(t) + ∫}{}_0{}^{\mathrm{t}}\text{a(t ? s) Au(s) ds ? F(t) (0 < t < ∞) (′ =}\text{d}\text{dt}\text{), u(0) = u}{}_0$, (¹) as well as the corresponding problem with infinite delay $\text{u′(t) + Bu(t) + ∫}{}_{\mathrm{?\infty }}{}^{\mathrm{t}}\text{a(t ? s) Au(s) ds ? ?(t) (0 < t < ∞), u(t) = h(t) (?∞ < t ? 0)}$. (⁷) Under various assumptions on the nonlinear operators A, B and on the given functions a, F, f, h existence theorems are obtained for (¹) and (⁷, followed by results concerning boundedness and asymptotic behaviour of solutions on (0 ? < ∞); two applications of the theory to problems of nonlinear heat flow with “infinite memory” are also discussed.     

Representation of natural numbers by sums of four squares of integers having a special form

This paper obtains an asymptotic formula for the number of solutions to the equation $ l_1^2 + { }l_2^2 + l_3^2 + l_4^2 = N $ in integers l ₁, l ₂, l ₃, l ₄ such that a < {??l _j } < b, where ?? is a quadratic irrational number, 0 ?? a < b ?? 1, j = 1, 2, 3, 4.     

Oscillation theorems for fourth order functional differential equations

The authors investigate the oscillatory behavior of all solutions of the fourth order functional differential equations $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})+q(t)f(x[g(t)])=0$ and $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})=q(t)f(x[g(t)])+p(t)h(x[\sigma(t)])$ in the case where ∫ ^∞ a ^?1/α (s)ds<∞. The results are illustrated with examples.     

Sequential conjugate-gradient-restoration algorithm for optimal control problems. Part 1. Theory

This paper considers the problem of minimizing a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the state and the parameter are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. First, the case of a quadratic functional subject to linear constraints is considered, and a conjugate-gradient algorithm is derived. Nominal functionsx(t),u(t), π satisfying all the differential equations and boundary conditions are assumed. Variations Δx(t), δu(t), Δπ are determined so that the value of the functional is decreased. These variations are obtained by minimizing the first-order change of the functional subject to the differential equations, the boundary conditions, and a quadratic constraint on the variations of the control and the parameter. Next, the more general case of a nonquadratic functional subject to nonlinear constraints is considered. The algorithm derived for the linear-quadratic case is employed with one modification: a restoration phase is inserted between any two successive conjugate-gradient phases. In the restoration phase, variations Δx(t), Δu(t), Δπ are determined by requiring the least-square change of the control and the parameter subject to the linearized differential equations and the linearized boundary conditions. Thus, a sequential conjugate-gradient-restoration algorithm is constructed in such a way that the differential equations and the boundary conditions are satisfied at the end of each complete conjugate-gradient-restoration cycle. Several numerical examples illustrating the theory of this paper are given in Part 2 (see Ref. 1). These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper. This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185. The authors are indebted to Professor A. Miele for stimulating discussions. Formerly, Graduate Studient in Aero-Astronautics, Department of Mechanical and Aerospace Engineering and Materials Science, Rice University, Houston, Texas.     

Fractional Integral Representation of Master Equation

Using the definition of Liouville–Riemann (L–R) fractional integral operator, master equation can be represented in the domain of fractal time evolution with a critical exponent a (0<a⩽1) . The relation between the continuous time random walks (CTRW) and fractional master equation (FME) has been achieved by obtaining the corresponding waiting time density (WTD) ψ (t) . The latter is obtained in a closed form in terms of the generalized Mittag–Leffler (M–L) function. The asymptotic expansion of the (M–L) function show the same behavior considered in the theory of random walk. Applying the Fourier and Laplace–Mellin transforms to (FME) , one obtains the solution, in closed form, in terms of the Fox function.     

The Fuller index and global Hopf bifurcation

Using an index for periodic solutions of an autonomous equation defined by Fuller, we prove Alexander and Yorke's global Hopf bifurcation theorem. As the Fuller index can be defined for retarded functional differential equations, the global bifurcation theorem can also be proved in this case. These results imply the existence of periodic solutions for delay equations with several rationally related delays, for example, $\text{x}\text{?}\text{(t) = ?α[ax(t ? 1) + bx(t ? 2)]g(x(t))}$, with a and b non-negative and α greater than some computable quantity ξ(a, b) calculated from the linearized equation.     

On nonlinear parabolic variational inequalities containing nonlocal terms

We investigate nonlinear parabolic variational inequalities which contain functional dependence on the unknown function. Such parabolic functional differential equations were studied e.g. by L. Simon in [8] (which was motivated by the work of M. Chipot and L. Molinet in [4]), where the following equation was considered: (1) $$ \begin{array}{*{20}c} {D_t u(t,x) - \sum\limits_{i = 1}^n {D_i \left[ {a_i (t,x,u(t,x),Du(t,x);u)} \right]} } \\ { + a_0 (t,x,u(t,x),Du(t,x);u) = f(t,x)} \\ {(t,x) \in Q_T = (0,T) \times \Omega ,a_i :Q_T \times R^{n + 1} \times L^p (0,T;V) \to R,} \\ \end{array} $$ where V denotes a closed linear subspace of the Sobolev-space W ^1,p (Ω) (2 ≦ p < ∞). In the above mentioned paper existence of weak solutions of the above equation is shown. These results were extended to systems of functional differential equations in [2]. In the following, we extend these existence results to variational inequalities by using the (less known) results of [6]. Finally, we show some examples.