Generalized evolution variational inequalities and their applications to partial differential equations

We study the existence of solutions to generalized evolution vari ational inequalities by pseudomonotone mapping theory. We obtain results which generalize and extend previously known theorems. We then apply our results to parabolic partial differential equations with discontinuous nonlinearities.     

Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay

This paper presents retarded integral inequalities of Henry-Gronwall type. Applying these inequalities, we study certain properties of solutions to fractional differential equations with delay.     

Lyapunov inequalities for partial differential equations

This paper is devoted to the study of Lp Lyapunov-type inequalities (1?p?+∞) for linear partial differential equations. More precisely, we treat the case of Neumann boundary conditions on bounded and regular domains in RN. It is proved that the relation between the quantities p and N/2 plays a crucial role. This fact shows a deep difference with respect to the ordinary case. The linear study is combined with Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear problems.     

Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations

Some new weakly singular integral inequalities of Gronwall-Bellman type are established, which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations, integral equations and evolution equations. Some applications to fractional differential and integral equations are also indicated.     

Nonlinear fractional differential equations of Sobolev type

Sobolev type nonlinear equations with time fractional derivatives are considered. Using the test function method, limiting exponents for nonexistence of solutions are found. Copyright © 2013 John Wiley & Sons, Ltd.     

Spatial discretization of partial differential equations with integrals

We consider the problem of constructing spatial finite-differenceapproximations on an arbitrary fixed grid which preserve anynumber of integrals of the partial differential equation andpreserve some of its symmetries. A basis for the space of suchfinite-difference operators is constructed; most cases of interestinvolve a single such basis element. (The ‘Arakawa’Jacobian is such an element, as are discretizations satisfying‘summation by parts’ identities.) We show how thegrid, its symmetries, and the differential operator interactto affect the complexity of the finite difference.     

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the 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. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.     

A fractional characteristic method for solving fractional partial differential equations

The method of characteristics has played a very important role in mathematical physics. Previously, it has been employed to solve the initial value problem for partial differential equations of first order. In this work, we propose a new fractional characteristic method and use it to solve some fractional partial differential equations.     

General methods for converting impulsive fractional differential equations to integral equations and applications

In this paper, we propose the concepts of Caputo fractional derivatives and Caputo type Hadamard fractional derivatives for piecewise continuous functions. We obtain general solutions of four classes of impulsive fractional differential equations (Theorem 3.1–Theorem 3.4) respectively. These results are applied to converting boundary value problems for impulsive fractional differential equations to integral equations. Some comments are made on recently published papers (see Section 4).     

Control problems governed by Sobolev partial differential equations

Let G be a bounded subset of Rⁿ with a smooth boundary and Q = G × (0, T]. We consider a control problem governed by the Sobolev initial-value problem My_t(u) + Ly(u) = u in L²(Q), y(·, 0; u) = 0 in L₂(G), where M = M(x) and L = L(x) are symmetric uniformly strongly elliptic operators of orders 2m and 2l, respectively. The problem is to find the control u₀ of L²(Q)-norm at most b that steers to within a prescribed tolerance ? of a given function Z in L²(G) and that minimizes a certain energy functional. Our main results establish regularity properties of u₀. We also give results concerning the existence and uniqueness of the optimal control, the controllability of Sobolev initial-value problems, and properties of the Lagrange multipliers associated with the problem constraints.     

On refined Hardy-type inequalities with fractional integrals and fractional derivatives

The aim of this paper is to present new more general Hardy-type inequalities for different kinds of fractional integrals and fractional derivatives.     

Coerciveness inequalities for nonelliptic systems of partial differential equations

Annali di Matematica Pura ed Applicata (1923 -) - This paper deals with first-order matrix partial differential operators of the form (1) $$L = - i \mathop \sum \limits_{j = 1}^n L_j (x)D_i + L_o...     

Solvability of hyperbolic fractional partial differential equations

The main purpose of this paper is to study the existence and uniqueness of solutions for the hyperbolic fractional differential equations with integral conditions. Under suitable assumptions, the results are established by using an energy integral method which is based on constructing an appropriate multiplier. Further we find the solution of the hyperbolic fractional differential equations using Adomian decomposition method. Examples are provided to illustrate the theory.     

Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations

Some inequalities concerning the Itô stochastic integral and solutions of stochastic different equations are obtained.     

RBFs approximation method for time fractional partial differential equations

In this paper, radial basis functions (RBFs) approximation method is implemented for time fractional advection–diffusion equation on a bounded domain. In this method the first order time derivative is replaced by the Caputo fractional derivative of order α ∈ (0, 1], and spatial derivatives are approximated by the derivative of interpolation in the Kansa method. Stability and convergence of the method is discussed. Several numerical examples are include to demonstrate effectiveness and accuracy of the method.     

Euler integrals and multi-integrals of linear partial differential equations

In this paper, we discuss the notion of reducibility of solutions of the Euler form generalizing solutions arising from the Laplace cascade method for integrating second-order hyperbolic equations in the plane. We present reduction algorithms and prove the equivalence of various possible exact definitions of the reduction of similar explicit solutions.     

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E:

(SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.