Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel

An integro-differential equation involving a convolution integral with a weakly singular kernel is considered. The kernel can be that of a fractional integral. The integro-differential equation is discretized using the discontinuous Galerkin method with piecewise constant basis functions. Sparse quadrature is introduced for the convolution term to overcome the problem with the growing amount of data that has to be stored and used in each time-step. A priori and a posteriori error estimates are proved. An adaptive strategy based on the a posteriori error estimate is developed. Finally, the precision and effectiveness of the algorithm are demonstrated in the case that the convolution is a fractional integral. This is done by comparing the numerical solutions with analytical solutions.

[1]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[2]  Peter J. Torvik,et al.  Fractional calculus in the transient analysis of viscoelastically damped structures , 1983 .

[3]  K. Diethelm.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

[4]  Vidar Thomée,et al.  Numerical solution of an evolution equation with a positive-type memory term , 1993, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[5]  Neville J. Ford,et al.  The numerical solution of fractional differential equations: Speed versus accuracy , 2001, Numerical Algorithms.

[6]  Vidar Thomée,et al.  Discretization with variable time steps of an evolution equation with a positive-type memory term , 1996 .

[7]  Stig Larsson,et al.  Adaptive discretization of fractional order viscoelasticity using sparse time history , 2004 .

[8]  N. Ford,et al.  Numerical Solution of the Bagley-Torvik Equation , 2002, BIT Numerical Mathematics.

[9]  M. Enelund,et al.  Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws , 1999 .

[10]  W. E. Scott.  Operational calculus based on the two-sided Laplace integral , 1951 .

[11]  T. Nonnenmacher,et al.  Fractional integral operators and Fox functions in the theory of viscoelasticity , 1991 .

[12]  Vidar Thomée,et al.  Time discretization of an integro-differential equation of parabolic type , 1986 .

[13]  F. Oberhettinger,et al.  Tables of Laplace Transforms , 1973 .

[14]  Nai Ying Zhang,et al.  On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type , 1993 .

[15]  C. Lubich.  Discretized fractional calculus , 1986 .

[16]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[17]  M. Caputo,et al.  A new dissipation model based on memory mechanism , 1971 .

[18]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .