Orthogonal cubic spline basis and its applications to a partial integro-differential equation with a weakly singular kernel

In this paper, a new orthogonal basis for the space of cubic splines has been introduced. This basis is obtained based on B-splines and using the Gram–Schmidt orthogonalization process. A linear combination of these basis is used to estimate functions and numerical solutions of a partial integro-differential equation with a weakly singular kernel. The convergence analysis in the approximate scheme is investigated. The accuracy of the proposed method is demonstrated by three test problems. The results of numerical experiments are compared with analytical solutions by calculating errors $$L_2$$ -norm, $$L_\infty $$ -norm and $$H^1$$ -norm. The numerical results are found to be in good agreement with the exact solutions.

[1]  Yubin Yan,et al.  High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problem , 2020, Mathematical Methods in the Applied Sciences.

[2]  J. Biazar,et al.  FD-RBF for Partial Integro-Differential Equations with a Weakly Singular Kernel , 2015 .

[3]  Hiroshi Fujiwara,et al.  High-Accurate Numerical Method for Integral Equations of the First Kind under Multiple-Precision Arithmetic , 2003 .

[4]  Jafar Saberi-Nadjafi,et al.  Cubic B-splines collocation method for solving a partial integro-differential equation with a weakly singular kernel , 2019 .

[5]  Gamal N. Elnagar,et al.  Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations , 1996 .

[6]  Haixiang Zhang,et al.  The BDF orthogonal spline collocation method for the two-dimensional evolution equation with memory , 2018, Int. J. Comput. Math..

[7]  R. Christensen,et al.  Theory of Viscoelasticity , 1971 .

[8]  Farshid Mirzaee,et al.  Cubic B-spline approximation for linear stochastic integro-differential equation of fractional order , 2020, J. Comput. Appl. Math..

[9]  Da Xu,et al.  Quasi wavelet based numerical method for a class of partial integro-differential equation , 2012, Appl. Math. Comput..

[10]  M. Gurtin,et al.  A general theory of heat conduction with finite wave speeds , 1968 .

[11]  M. Aguilar,et al.  Collocation methods for second-order Volterra integro-differential equations , 1988 .

[12]  Qiong Tang,et al.  Discrete-time orthogonal spline collocation method for a modified anomalous diffusion equation , 2021, Int. J. Comput. Math..

[13]  Hermann Brunner,et al.  Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution , 1997 .

[14]  Fawang Liu,et al.  An implicit RBF meshless approach for time fractional diffusion equations , 2011 .

[15]  MEHDI DEHGHAN,et al.  Solution of a partial integro-differential equation arising from viscoelasticity , 2006, Int. J. Comput. Math..

[16]  Tao Tang,et al.  A finite difference scheme for partial integro-differential equations with a weakly singular kernel , 1993 .

[17]  J. G. Verwer,et al.  Solving parabolic integro-differential equations by an explicit integration method , 1992 .

[18]  Tang,et al.  ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS , 2008 .

[19]  Akbar Mohebbi,et al.  Compact finite difference scheme for the solution of a time fractional partial integro‐differential equation with a weakly singular kernel , 2017 .

[20]  J. C. Mason,et al.  Orthogonal splines based on B-splines — with applications to least squares, smoothing and regularisation problems , 2005, Numerical Algorithms.

[21]  Da Xu,et al.  An alternating direction implicit orthogonal spline collocation method for the two dimensional multi-term time fractional integro-differential equation , 2020 .

[22]  Michael Renardy,et al.  Mathematical Analysis of Viscoelastic Flows , 1987 .

[23]  Suheil A. Khuri,et al.  A spline collocation approach for a generalized wave equation subject to non-local conservation condition , 2010, Appl. Math. Comput..

[24]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[26]  H. Brunner.  Implicit Runge-Kutta-Nyström methods for general second-order Volterra integro-differential equations , 1987 .

[27]  G. Fairweather,et al.  Finite element methods for parabolic and hyperbolic partial integro-differential equations , 1988 .

[28]  Jalil Rashidinia,et al.  Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method , 2017, Advances in Difference Equations.

[29]  Da Xu,et al.  A compact difference scheme for a partial integro-differential equation with a weakly singular kernel , 2015 .