Abstract
In this paper we consider a particular class of two-dimensional singular Volterra integral equations. Firstly we show that these integral equations can indeed arise in practice by considering a diffusion problem with an output flux which is nonlocal in time; this problem is shown to admit an analytic solution in the form of an integral. More crucially, the problem can be re-characterized as an integral equation of this particular class. This example then provides motivation for a more general study: an analytic solution is obtained for the case when the kernel and the forcing function are both unity. This analytic solution, in the form of a series solution, is a variant of the Mittag-Leffler function. As a consequence it is an entire function. A Gronwall lemma is obtained. This then permits a general existence and uniqueness theorem to be proved.
| Original language | English |
|---|---|
| Pages (from-to) | 243-252 |
| Number of pages | 10 |
| Journal | Journal of Mathematical Analysis and Applications |
| Volume | 423 |
| Issue number | 1 |
| Early online date | 2 Oct 2014 |
| DOIs | |
| Publication status | Published - 1 Mar 2015 |
Funding
The first author would like to acknowledge a Royal Society of Edinburgh Grant and thank his surgeon, Professor Leung, without whom this paper would not have been written. Both authors acknowledge FAPESP support through Grant No. 2013/07375-0 . Appendix A From the Hankel contour (see ) we observe that Fig. 1 1 2 π i ∫ γ − i ∞ γ + i ∞ e s t c ¯ ( 1 , s ) d s = − ∑ H k 1 2 π i ∫ H k e s t c ¯ ( 1 , s ) d s where the sum is over the segments AB and DE . This is evident since the integrals around the circle of radius R ( R → ∞ ) and around the circle BCD of radius ϵ ( ϵ → 0 ) are both zero. Thus c ( 1 , t ) = L − 1 [ e s t c 0 s + coth s ] = − c 0 2 π i { ∫ A B e s t s + coth s d s + ∫ D E e s t s + coth s d s } = − c 0 2 π i ( I 1 + I 2 ) . Consider I 1 and write s = x e i π = − x and hence s = x e i π / 2 = i x . Thus I 1 = lim ϵ → 0 R → ∞ ∫ − R − ϵ e s t s + coth s d s = − ∫ 0 ∞ e − x t ( x − i cot x ) d x . Consider I 2 and write s = x e − i π = − x , s = x e − i π / 2 = − i x . Thus, similarly I 2 = ∫ 0 ∞ e − x t x + i cot x d x . So c ( 1 , t ) = − c 0 2 π i [ − ∫ 0 ∞ e − x t ( x − i cot x ) d x + ∫ 0 ∞ e − x t ( x + i cot x ) d x ] = c 0 π ∫ 0 ∞ e − x t cot x x 2 + cot 2 x d x , or equivalently, c ( 1 , t ) = c 0 π ∫ 0 ∞ e − x t sin x cos x x 2 sin 2 x + cos 2 x d x .
Keywords
- Mittag-Leffler function
- non-local diffusion
- two-dimensional Volterra integral equation