Volume 30, Issue 1

In this paper a new direct proof for the irrationality of Euler´s number   e =∑ k =0 ∞ 1/ k !   is presented. Furthermore, formulas for the base b digits are given which, however, are not computably effective. Finally we generalize our method and give a simple criterium for some fast converging series representing irrational numbers.

Our paper is devoted to investigation of partial differential equations of fractional order. We give a historical survey of results in this field basically concerning differential equations with Riemann–Liouville and Caputo partial fractional derivatives. We pay a special attention to application of the method of Fourier, Laplace and Mellin integral transforms to study partial fractional differential equations. We present recent results on explicit solutions of Cauchy-type and Cauchy problems for model homogeneous partial differential equations with Riemann–Liouville and Caputo partial fractional derivatives generalizing the classical heat and wave equations. Explicit solutions of the above problems are given in terms of the Mittag–Leffler function, and of the so-called H -function and its special cases such as the Wright and generalized Wright functions. We discuss applications of partial fractional differential equations to the modelling of anomalous phenomena in nature and in the theory of complex systems. Special interest has been paid to the anomalous diffusion processes such as super-slow diffusion (or sub-diffusion) and super-fast diffusion (or super-diffusion) processes. We present new fractional derivative model, which allows us to have strong control of both the sub- and super-diffusion processes which means the control of the temporal behavior of the speed of spreading via analysis of the second space moment or moments of some other order.

In this paper, we consider two bounded open subsets of Ω i and Ω o of R n containing 0 and a (nonlinear) function G o of ∂Ω o ×R n to R n , and a map T of ]1-(2/ n ),+∞[ times the set M n (R) of n × n matrices with real entries to M n (R), and we consider the problem   div ( T ( ω , Du ))=0  in Ω o ε clΩ i ,   {- T ( ω , Du ) ν ε Ω i =0 on ε ∂Ω i ,   (ω , Du ( x )) ν o ( x )= G o ( x , u ( x )) ∀ x ∈∂Ω o ,   where ν ε Ω i and ν o denote the outward unit normal to ε ∂Ω i and ∂Ω o , respectively, and where ε >0 is a small parameter. Here ( ω -1) plays the role of ratio between the first and second Lamé constants, and T ( ω ,·) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor, and G o plays the role of (a constant multiple of) a traction applied on the points of ∂Ω o . Then we prove that under suitable assumptions the above problem has a family of solutions { u ( ε ,·)} ε ∈]0, ε ´[ for ε ´ sufficiently small and we show that in a certain sense { u ( ε ,·)} ε ∈]0, ε ´[ can be continued real analytically for negative values of ε .

Several problems of plane elasticity theory which can be reduced to different biharmonic boundary value problems are described, and the concept of biharmonic Green functions is discussed.   The explicit formula for the particular biharmonic Green function in a circular ring   R = { z ∈ C: 0< r <| z |<1}   is presented and applied to the solution of the Dirichlet boundary value problem for the bi-Poisson equation   (∂ z ∂ z¯ ) 2 w = f in R , w = γ 0 , ;∂ z ∂ z¯ w = γ 1 on ∂ R ,   for given f ∈ L p ( R ;C), p >2, γ 0 , γ 1 ∈ C (∂ R ;C).