To get a smooth flux profile, i.e., a function with continuous first derivative, the degree of the spline function should be at least two, and for this reason the degree will be fixed to two in this work, i.e., k=2.
Rather than relying on ad hoc approximation methods, the dispersion integrals for an arbitrary distribution function with continuous derivative are systematically expanded in terms of a set of orthogonal functions for which the corresponding dispersion functions are known.
And F is a function with continuous first and second (partial) derivatives with respect to all its arguments.
Let (F t,x_{1},ldots,x_{n}, y_{1},ldots,y_{2n})) be a function with continuous first and second (partial) derivatives with respect to all its arguments.
Assume that (F=F t,x,D_{t_{0},t}^{alpha}x,D_{t,{t_{f}}}^{beta}x)) is a function with continuous first and second (partial) derivatives with respect to all its arguments.
For any (n in N), have (mathit {AC}^{n}[0, 1]) to be the space consisting of all functions (u(t)) with continuous derivatives until (n-1) order on a closed interval ([0, 1]) such that (u^{n-1}(t)) is absolutely continuous, which we denote mathit {AC}^{n}[0, 1]= bigl{ umid[0,1]rightarrowmathbb{R}text{ and }D^{n-1}u(t text{ is absolutely continuous in } [0,1] bigr}.
C k denotes the space of functions on R d with continuous derivatives up to order k.
Theorem 1 Let F ( x ) = F ( x 1, …, x n ) be a symmetric function with continuous partial derivatives on I n = I × I × ⋯ × I, where I is an open interval.
However, our result holds in a more general setting, that is, for differentiable functions with absolutely continuous derivatives, in a measure-theoretic (and probabilistic) form.
The space of all continuous real functions on I with values in (mathbb{R}) and the space of all functions with two continuous derivatives on the interval I are shown by the standard notations (C I,mathbb{R})) and (C^{2}(I,mathbb{R})), respectively.
Consider the space (AC^{n}[0,T]) of functions with absolutely continuous derivatives up to order (n-1) and with absolutely continuous n-derivative.
