Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator

This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function.     

An extension of the Garrett-Stanojevic class

The extension is made for the theorem of [2], by considering the classes S_pr and C_r, 1<p≤2, r∈{0,1,2,…} instead of S_p and C. Namely, it is shown that the class S_pr is a subclass of C_r∩BV, 1<p≤2,r∈{0,1, 2,…}, where BV is the class of null sequences of bounded variation, and C_r, r∈{0,1,2…} is the extension of the Garrett-Stanojevci class.     

Some Classes of L 1-Convergence of Fourier Series

In this paper equivalent classes of the classes M and S _pr , p > 1, > 0. r { 0,1,2, ... ,[]} defined by Shuyun [3] are obtained. Then, it is shown that the class S _pr , 1 > p 2, 0, r {0,1,2,...,[]} is a subclass of BVC _r , where S _pr is the equivalent class of the Shuyun's class S _pr , BV is the class of null sequences of bounded variation and C _r is the extension of the Garrett--Stanojevic class. As a corollary of this result, we have obtained the theorem, proved in [7].     

Remarks on some classes of Fourier coefficients

In this paper equivalent classes of the classes M' and S' _{p r}, p >1, 0,r {0,1,2,...,[]} defined by Sheng [5] are obtained. Then it is shown that the classes of Fourier coefficients S _p, S' _p(case r==0) and S _p(), p>1, defined by . V. Stanojevi, V. B. Stanojevi Sheng and the author of the present note are identical. As a corollary of this result, the L ¹-estimate for cosine series, obtained in [10], is refined.     

An extension of the Garrett-Stanojevic class

The extension is made for the theorem of [2], by considering the classes S_pr and C_r, 1<p≤2, r∈{0,1,2,…} instead of S_p and C. Namely, it is shown that the class S_pr is a subclass of C_r∩BV, 1<p≤2,r∈{0,1, 2,…}, where BV is the class of null sequences of bounded variation, and C_r, r∈{0,1,2…} is the extension of the Garrett-Stanojevci class.     

Modeling of a voltammetric experiment in a limiting diffusion space

A voltammetric experiment confined in a limiting diffusion space is analyzed theoretically governed by conventional or time-anomalous factional diffusion under conditions of cyclic and square-wave voltammetry. The solution for conventional diffusion is derived by means of the Jacobi theta function Q( a²/p²t )( a = LD ^{- 1/2} \Theta \left( {{a^2}/{\pi^2}t} \right)\left( {a = L{D^{ - 1/2}}} \right. , where L is the thickness of the finite diffusion space, D is the diffusion coefficient, and t is the time of the experiment) and compared with the solution frequently used in the literature expressed in the form Θ(a ⁻² t). For L → ∞, the present solution converges to the one for the semi-infinite diffusion, thus being of a general applicability for both finite and semi-infinite diffusion. Hence, the mathematical model for simulation of both cyclic and square-wave voltammetric experiment provides significant advances in terms of simulation time and accuracy compared to the previous model based on the modified step-function method Mirčeski (J Phys Chem B 108:13719, 2004). For the fractional diffusion experiment, the solution is derived by combining an infinite series and the Wright function f( - a/2,a/2; - 2ax ^{- 1/2}t ^{- a/2} ) \phi \left( { - \alpha /2,\alpha /2; - 2a{\xi^{ - 1/2}}{t^{ - \alpha /2}}} \right) , where α is the time fractional parameter ranging over the interval 0 < a < 1 0 < \alpha < {1} , and ξ = 1 s^1−α is the auxiliary constant. The voltammetric properties of the experiment controlled by fractional diffusion are comparable for both finite and semi-infinite diffusion.     

Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions

We give an analytical treatment of a time fractional diffusion equation with Caputo time-fractional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experiment in limiting diffusion space.