《走出去：Fractional Partial Differential Equations and their Numerical Solutions》：
　　Chapter 1
　　Physics Background
　　Fractional differential equations have profound physical backgrounds and rich related theories， and are noticeable in recent years. They are equations containing fractional derivative or fractional integrals， which have applied in various disciplines such as physics， biology and chemistry. More specifically， they are widely used in dynamical systems with chaotic dynamical behavior， quasi-chaotic dynamical systems， dynamics of complex material or porous media and random walks with memory. The purpose of this chapter is to introduce the origin of the fractional derivative， and then some physical backgrounds of fractional differential equations. Due to space limitations， this chapter only gives some brief introductions. Even so， these are sufficient to show that the fractional differential equations， including fractional partial differential equations and fractional integral equations， are widely employed in various applied fields. However， further mathematical theories and numerical algorithms of fractional differential equations need to be studied. Interested readers can refer to more monographs and literatures.
　　1.1 Origin of the fractional derivative
　　The concepts of integer order derivative and integral are well known. The derivative dny/dxn describes the changes of variable y with respect to variable x， supported by profound physical backgrounds. Now the problem is how to generalize n into a fraction， even a complex number.
　　The long-standing problem can be traced back to the letter from L’H.opital to Leibniz in 1695， in which it is asked like what the derivative dny/dxn is when n = 1/2. In the same year， the derivative of general order was mentioned in the letter from Leibniz to J. Bernoulli as well. The problem was also considered by Euler（1730）， Lagrange（1849） et al， who gave some relevant insights. In 1812， by using the concept of integral， Laplace provided a definition of fractional derivative. When y = xm， using the gamma function， was derived by Lacroix， who gives . This is consistent with the Riemann-Liouville fractional derivative in the present.
　　Soon later， Fourier （1822） gave the definition of fractional derivative through the Fourier transform. The function f（x） can be expressed as a double integral By replacing n with ν， and calculating the derivative under the integral sign， one can generalize the integer order derivative into the fractional order derivative Consider the Abel integral equation where f is to be determined. The right-hand side defines a definite integral of fractional integral with order 1/2. Abel wrote √π dx.1/2 f（x） for the righthand
　　component， then ， which indicates that the fractional derivative of a constant is no longer zero.
　　In 1930s， Liouville， possibly inspired by Fourier and Abel， made a series of work in the field of fractional derivative， and successfully applied them into the potential theory. Since Dmeax = ameax，
　　1.1 Origin of the fractional derivative 3
　　the order of the derivative was generalized to be arbitrary by Liouville （ν can be a rational number， an irrational number， even a complex number） If a function f can be expanded into an infinite series its fractional derivative can be obtained as How can we obtain the fractional derivative if f cannot be written in the form of equation （1.1.5）? Liouville probably had noticed this problem， and he gave another expression by using the Gamma function. In order to make use of the basic assumptions （1.1.4）， noting that one then obtains So far， we have introduced two different definitions of fractional derivatives.
　　One is the definition （1.1.1） with respect to xa（a > 0） given by Lacroix， the other is the definition （1.1.7） with regard to x.a（a > 0） given by Liouville.
　　It can be seen that， Lacroix’s definition shows that the fractional derivative of a constant x0 is no longer zero. For instance， when m = 0， n = However， in Liouville’s definition， since Γ（0） = ∞， the fractional derivative of a constant is zero （despite Liouville’s assumption a > 0）. As far as which of the two definitions is the correct form of fractional derivative， Willian Center pointed out it can be attributed to how to determine dνx0/dxν； and as DeMorgan indicated （1840）， both of them may very possibly be parts of a more general system.
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