We consider the homogeneous equation Au = 0, where A is a symmetric and coercive elliptic operator in H1(Ω) with Ω bounded domain in ℝd. The boundary conditions involve fractional power α, 0 < α < 1, of the Steklov spectral operator arising in Dirichlet to Neumann map. For such problems we discuss two different numerical methods: (1) a computational algorithm based on an approximation of the integral representation of the fractional power of the operator and (2) numerical technique involving an auxiliary Cauchy problem for an ultra-parabolic equation and its subsequent approximation by a time stepping technique. For both methods we present numerical experiment for a model two-dimensional problem that demonstrate the accuracy, efficiency, and stability of the algorithms.
L. Aceto and P. Novati, Rational approximation to the fractional Laplacian operator in reaction-diffusion problems. SIAM J. Scientific Computing39, No 1 (2017), A214–A228; doi: 10.1137/16M1064714.
M.G. Armentano, The effect of reduced integration in the Steklov eigenvalue problem. ESAIM: Mathematical Modelling and Numerical Analysis38, No 1 (2004), 27–36; doi: 10.1051/m2an:2004002.
I. Babuska and J. Osborn, Eigenvalue problems. In: Handbook of Numerical Analysis Vol. 2, North-Holland, Amsterdam (1991), 641–787.
A. Bonito and J. Pasciak, Numerical approximation of fractional powers of elliptic operators. Mathematics of Computation84, No 295 (2015), 2083–2110; DOI: 10.1090/S0025-5718-2015-02937-8.
A. Bueno-Orovio, D. Kay, and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numerical Mathematics54, No 4 (2014), 1–18; doi:10.1007/s10543-014-0484-2.
K. Burrage, N. Hale, and D. Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM Journal on Scientific Computing34, No 4 (2012), A2145–A2172; DOI:10.1137/110847007.
I. Gavrilyuk, W. Hackbusch, and B. Khoromskij, Data-sparse approximation to the operator-valued functions of elliptic operator. Mathematics of Computation73, No 247 (2004), 1297–1324; doi: http://www.jstor.org/stable/4099897.
I. Gavrilyuk, W. Hackbusch, and B. Khoromskij, Data-sparse approximation to a class of operator-valued functions. Mathematics of Computation74, No 250 (2005), 681–708; doi: 10.1090/S0025-5718-04-01703-X.
S. Harizanov, R. Lazarov, P. Marinov, S. Margenov, and Y. Vutov, Optimal solvers for linear systems with fractional powers of sparse SPD matrices. Submitted to: Numerical Linear Algebra with Applications, posted as arXiv:1612.04846v1.
N.J. Higham, Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008).
M. Ilić, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation.I. Fract. Calc. Appl. Anal.8, No 3 (2005), 323–341; at http://www.math.bas.bg/∼fcaa.
M. Ilić, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation, II. With nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal.9, No 4 (2006), 333–349; at http://www.math.bas.bg/∼fcaa.
M. Ilić, I.W. Turner, and V. Anh, A numerical solution using an adaptively preconditioned Lanczos method for a class of linear systems related with the fractional Poisson equation. Intern. J. of Stochastic Analysis (2008), 1–26, Article ID 104525.
A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies, Elsevier, Amsterdam (2006).
M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik, and P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions. Noordhoff International Publishing (1976).
R. Metzler, J.H. Jeon, A.G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Physical Chemistry Chemical Physics16, No 44 (2014), 24128–24164; doi: 10.1039/c4cp03465a.
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, Vol. 198, Academic Press (1998).
A.A. Samarskii, The Theory of Difference Schemes. Marcel Dekker, New York (2001).
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Ser. in Computational Mathematics, Vol. 25, Springer (2006).
P.N. Vabishchevich, Numerically solving an equation for fractional powers of elliptic operators. Journal of Computational Physics282, No 1 (2015), 289–302; doi: 10.1016/j.jcp.2014.11.02.
Fractional Calculus and Applied Analysis (FCAA) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order.