Rothe’s method for solving some fractional integral diffusion equation Abdur Raheem ⇑ , Dhirendra Bahuguna Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India article info Keywords: Fractional integral equation Diffusion equation Strong solution Semigroup of bounded linear operators Method of semidiscretization abstract In this paper, we apply the Rothe’s method to a fractional integral diffusion equation and establish the existence and uniqueness of a strong solution. As an application, we include an example to illustrate the main result. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction In this paper we apply the Rothe’s method to the following fractional integral diffusion equation in a Banach space X @uðtÞ @t þ AuðtÞ¼ 1 CðaÞ Z t 0 uðsÞ ðt  sÞ 1a ds þ f ðtÞ; t 2ð0; T ; ð1Þ uð0Þ¼ u 0 ; ð2Þ where 0 < a < 1; A is the infinitesimal generator of a C 0 -semigroup of contractions, f is a given map from ½0; T  into X; u 0 2 DðAÞ A, the domain of A. The problem considered in this paper is a particular case of the fractional integral diffusion problem D b uðtÞþ AuðtÞ¼ 1 CðaÞ Z t 0 uðsÞ ðt  sÞ 1a ds þ f ðtÞ; uð0Þ¼ u 0 ; where 0 < a 6 1; 0 < b 6 1. If we take b ¼ 1 and 0 < a < 1, then above problem reduces to the problem (1) and (2). In 1930, E. Rothe [8] has introduced a method to solve the following scalar parabolic initial boundary value problem of second order Rðt; xÞ @u @t  @ 2 u @x 2 ¼ Sðt; x; uÞ; 0 < x < 1; t > 0; uð0; xÞ¼ u 0 ðxÞ; uðt; 0Þ¼ uðt; 1Þ¼ 0; t P 0; http://dx.doi.org/10.1016/j.amc.2014.03.025 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: araheem.iitk3239@gmail.com (A. Raheem), dhiren@iitk.ac.in (D. Bahuguna). Applied Mathematics and Computation 236 (2014) 161–168 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc