Type: Article

Fractional derivative of Abel type on a half-line

Elena I. Kaikina;

Abstract:

Abstract We consider the initial-boundary value problem on a half-line for an evolution equation (partial derivative(t) + vertical bar partial derivative(x)vertical bar(alpha))u(x, t) = f(x, t), t > 0, x > 0, with a fractional derivative of Abel type (0.1) vertical bar partial derivative(x)vertical bar(alpha) u = R1-alpha+[alpha]partial derivative([alpha]+1)(x)u, where [alpha] denotes the integer part of number alpha > 0, alpha is not equal to an integer, and R(alpha)u = 1/2 Gamma(alpha)sin(pi/2 alpha) integral(+infinity)(0) sign(x - y)u(y)/vertical bar x - y vertical bar(1-alpha) dy is the modified Riesz potential. We study traditionally important problems of a theory of partial differential equations, such as existence and uniqueness of solution. We propose a new method of solution. Also we get a closed form of the solution.