x

The Fresnel cosine integral C(x), the Fresnel sine integral S(x), and the associated functions C

The Fresnel cosine integral C(x) is defined by (see [3]) and the associated functions C + (x) and C − (x) are defined by The Fresnel sine integral S(x) is defined by (see [3]) and the associated functions S + (x) and S − (x) are defined by where H denotes Heaviside's function.
We define the function I r (x) by for r = 0, 1, 2,....In particular, We define the functions cos + x, cos − x, sin + x, and sin − x by cos + x = H(x) cos x, cos − x = H(−x) cos x, If the classical convolution f * g of two functions f and g exists, then g * f exists and f * g = g * f . (8) Further, if (f * g) and f * g (or f * g) exist, then The classical definition of the convolution can be extended to define the convolution f * g of two distributions f and g in Ᏸ with the following definition, see [2].Definition 1.Let f and g be distributions in Ᏸ .Then the convolution f * g is defined by the equation for arbitrary ϕ in Ᏸ , provided that f and g satisfy either of the conditions (a) either f or g has bounded support, (b) the supports of f and g are bounded on the same side.It follows that if the convolution f * g exists by this definition, then (6) and (8) are satisfied.
Definition 1 was extended in [1] with the next definition but first of all we let τ be a function in Ᏸ having the following properties: Definition 6.Let f and g be distributions in Ᏸ and let f ν = f τ ν for ν > 0. The neutrix convolution product f * g is defined as the neutrix limit of the sequence {f ν * g}, provided that the limit h exists in the sense that for all ϕ in Ᏸ, where N is the neutrix, see van der Corput [5], with its domain N the positive real numbers, with negligible functions finite linear sums of the functions and all functions which converge to zero in the normal sense as ν tends to infinity.
Note that in this definition the convolution product f ν * g is defined in Gel'fand and Shilov's sense, since the distribution f ν has bounded support.
It was proved in [1] that if f * g exists in the classical sense or by Definition 1, then f * g exists and The following theorem was also proved in [1].
Theorem 7. Let f and g be distributions in Ᏸ and suppose that the neutrix convolution product f * g exists.Then the neutrix convolution product f * g exists and We need the following lemma.

Proof.
It is easily proved that and it follows from (6) and ( 28) that (27) hold when r = 0, since see Olver [4].We also have and it follows that Equations ( 27) now follow by induction.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: