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routing protocols can be very disruptive in a MANET environment Design of secure routing protocols (see 4) is one approach to solve this problem The other approach is to detect routing attacks using IDS The latter approach will be discussed extensively in this chapter Another unique challenge of MANET is the limited bandwidth Traditional IDS systems require a large amount of sensor information to be sent around for analysis, which is typically not a problem in large enterprise networks Transmitting large amounts of intrusion detection data is often not an acceptable solution for a MANET Data reduction, which involves compressing the captured data being transmitted among nodes, is then a required feature for IDS in a MANET environment In a MANET, nodes tend to move a lot and therefore the connectivity of nodes changes dramatically This makes it challenging to collect data from several sensors to a central entity for analysis and correlation For example, the node that is responsible for analysis and correlation may lose connectivity to most of the sensors and gain connectivity to several new nodes due to node movements A potential way of overcoming this challenge may be to develop a much more exible architecture for intrusion detection where there is a loose association between sensors and analysis/correlation components These constraints also rule out the use of a centralized architecture for intrusion detection In addition, the detection algorithms must be able to seamlessly adapt to topological changes due to node mobility and intermittent connectivity due to the wireless environment MANET networks are also much more dynamic and unpredictable because connectivity depends on the movements of nodes, terrain, changes in the mission (eg for a military application or a rst responder application), node failures, weather, and other factors As a result, it is dif cult to accurately characterize normal behavior Hence, it is often dif cult to distinguish malicious behavior from normal but unexpected events For example, a network may be seen connecting and disconnecting from the rest of the network This may be symptomatic of an attack but can also be due to the fact that a node is moving in and out of range of the network Existing detection tools (anomaly detection tools in particular) may not be effective in such an environment because they have been developed with a much more static and predictable environment in mind and cannot deal with the dynamism and unpredictability of MANETs Finally, unlike in wireline networks, nodes in an ad hoc network have limited energy Hence, often only computationally simple, energy-ef cient detection strategies can be used by such nodes The detection algorithms must also be distributed as communication with a central computing unit will consume signi cant energy and bandwidth All these differences with wireline networks motivate the design of detection strategies speci cally geared towards ad hoc networks We summarize this discussion in Table 51.

71 Introduction 711 Motivations We give some traditional example applications of fractional calculus and then we brie y point out the theoretical references 7111 Fields of application The modeling of certain physical phenomena, described as long memory, can be carried out by introducing integro-differentials terms with weakly singular kernels (ie, locally integrable but not necessarily continuous like t 1 when 0 < < 1) in the equations of the dynamics of materials This is very frequent, for example, in linear viscoelasticity with long memory, where a fractional stress-strain dynamic relation can be proposed: see [BAG 86] for viscoelasticity; [KOE 84, KOE 86] for a presentation a little more formalized; [BAG 91] for a rich and quite detailed example; [BAG 83a] for a modal analysis in forced mode or [BAG 85] for a modal analysis in transient state and nally, [CAP 76] for a modeling which utilizes an equation with partial derivatives with fractional derivative in time There are also applications for modeling in chemistry of polymers [BAG 83b] or for modeling of dynamics at the interface of fractal structures: see [LEM 90] for the applied physical aspect and [GIO 92] for the theoretical physical aspect

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Moreover, fractional derivatives can appear naturally when a dynamic phenomenon is strongly conditioned by the geometry of the problem: a simple, very instructive example is presented in [TOR 84] See in particular [CARP 97] for examples in continuum mechanics and [POD 99] for many applications in engineering sciences 7112 Theories A detailed historical overview of the theory of fractional derivatives is given in [OLD 74]; moreover, this work is undoubtedly one of the rst attempts to assemble scattered results Recently, a theoretical synthesis was proposed in [MIL 93], where certain algebraic aspects of fractional differential equations of rational order are completely developed In mathematics, the Russian work [SAM 87] is authoritative; it compiles a set of unique de nitions and theories Pseudo-differential operators are mentioned in 7 of [TAY 96] and the rst article on the concept of diffusive representation was, as far as we know, section 5 of [STA 94] During the last 10 years, a number of themes have developed: see, in particular, the book [MAT 98c] for a general theoretical framework [MON 98], and for several applications derived from them 712 Problems From a mathematical point of view, these integro-differential relations or convolutions with locally integrable kernels (or L1 , ie, absolutely integrable on loc any interval [a, b]) are not simple to treat: analytically, the singular character of kernel t 1 (with 0 < < 1) problematizes the use of theorems based on the regularity of the latter (in [DAUT 84a], for example, the kernels are always supposed to be continuous) Numerically, it is not simple to treat this singularity at the temporal origin (although that is a priori possible by carrying out an integration by parts, which arti cially increases the order of derivation of the unknown function, while keeping a convolution with a more regular kernel) In the theories mentioned in section 7112, several problems appear First, the de nition of fractional derivatives poses problems for orders higher than 1 (in particular, fractional derivatives do not commute, which is extremely awkward and, in addition, the composition of integration and fractional derivatives of the same order do not necessarily give the identity); this leads, in practice, to the use of rather strict calculations and a reintroduction a posteriori of the formal solution in the.

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