Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

Achievement sets and sum ranges with ideal supports

We introduce the notion of ideally supported achievement sets for a series of real numbers. We analize their complexity and topological properties. We compare the notion of ideal achievement sets with the notion of ideally supported sum range of real series, considered by Filip?w and Szuca. We complete Filip?w and Szuca characterization of ideal sum ranges, [R. Filip?w, P. Szuca, Rearrangement of conditionally convergent series on a small set, J. Math. Anal. Appl. 362 (2010), no. 1, 64-71.], and we obtain some generalization of Riemann?s Theorem.

Subsums of conditionally convergent series in finite dimensional spaces

An achievement set of a series is a set of all its subsums. We study the properties of achievement sets of conditionally convergent series in finite dimensional spaces. The purpose of the paper is to answer some of the open problems formulated in [10]. We obtain general result for series with harmonic-like coordinates, that is A((-1)n+1n-?1,..., (-1)n+1n-?d) = Rd for pairwise distinct numbers ?1,..., ?d ? (0,1]. For d = 2, ?1 = 1, ?2 = 1/2 this problem was stated in [10].

Alteration in the sum of alternating series by simple rearrangement

It is well known that the sum of an absolutely convergent series is invariant under rearrangement of its terms. On the other hand, a conditionally convergent series, that is one which converges but the sum of whose absolute values is unbounded, can be rearranged to have any sum whatsoever, or diverge in any desired manner (see for example [1, §44]). A simple examplS of a conditionally convergent series is the alternating harmonic series (AHS), . In [2], the following theorem on rearrangement of the AHS was proved:Theorem A: The AHS remains convergent under a simple rearrangement (i.e. the sub-sequence of its positive terms and the sub-sequence of its negative terms are in their original order) when p of its positive terms alternate throughout with q of its negative terms, and the alteration in sum is (p/q).