On a stochastic fractional partial differential equation with a fractional noise

We will prove the existence, uniqueness and regularity of the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. Moreover, the absolute continuity of the solution is also obtained.     

On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1$\alpha >1$ driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.     

On a nonlinear distributed order fractional differential equation

We study the existence and the uniqueness of mild and classical solutions for a class of equations of the form . Such equations arise in distributed derivatives models of viscoelasticity and system identification theory. We also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations.     

The decay rate for a fractional differential equation

We consider the fractional differential equation

     

On a universal differential equation for the analytic terms of C-superpositions on the real line

It is proved that every continuous function on the real line can be approximated uniformly (in the sense of a specific norm) by superpositions of analytic functions, which are solutions of a single universal differential equation. Every superposition is some function belonging to . This improves a former result of the author, from which the superpositions are known to be continuous.     

On the continuation of solutions of a fractional partial differential equation

We study the continuation of solutions of a fractional partial differential equation. We show that a solution can be uniquely continued into a domain that is uniquely determined by the boundary part supporting the initial conditions and outside which the continuation is no longer unique.     

Numerical approach to differential equations of fractional order

In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.     

A generalized Gronwall inequality and its application to a fractional differential equation

This paper presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation.     

Multi-term fractional differential equation in reflexive Banach space

In this paper, we investigate the solvability of a multi-term fractional differential equation in a reflexive Banach space relative to weak topology. A particular case of our main result is treated.     

On the solvability of a boundary value problem for a fractional differential equation

We obtain sufficient conditions for the existence and uniqueness of a solution of a boundary value problem for a differential equation that contains a mixed fractional derivative.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Partial differential equation approach to E 7

By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E₇ into a sum of irreducible submodules. This essentially gives a partial differential equation proof of a combinatorial identity on the dimensions of certain irreducible modules of E₇. We also determine two three-parameter families of irreducible submodules in the solution space of Cartan's well-known fourth-order E₇-invariant partial differential equation.     

Partial differential equation approach to F 4

Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to explicitly find all the singular vectors of the polynomial representation of the simple Lie algebra of type F ₄ over its 26-dimensional basic irreducible module, which also supplements a proof of the completeness of Brion’s abstractly described generators. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic polynomials with degree k ⩾ 2 is greater than or equal to 〚k/3〛 + 〚(k − 2)/3〛 + 2.