A class of developed schemes for parabolic integro-differential equations

References

• M. Asadzadeh and D. Rostamy, On multiscale variational and streamline diffusion schemes for a coupled nonlinear telegraph system, J. Comput. Theor. Transport. 45 (2016), pp. 509–527.
   Web of Science ®Google Scholar
• P. Bocher, H.D. Meyer and G.V. Berghe, Modified gregory formulae base on mixed interpolation, Int. J. Comput. Math. 52 (1993), pp. 109–122.
   Web of Science ®Google Scholar
• P. Bocher, H.D. Meyer and G.V. Berghe, On Gregory-and modified Gregory-type corrections to Newton-Cotes quadrature, J. Comput. Appl. Math. 50 (1994), pp. 145–158.
   Web of Science ®Google Scholar
• H. Brunner and J.D. Lambert, Stability of numerical methods for Volterra integro-differential equations, Computing 12 (1974), pp. 75–89.
   Web of Science ®Google Scholar
• H. Brunner, T. Tang and J. Zhang, Numerical blow-Up of nonlinear parabolic integro-differential equations on unbounded domain, J. Sci. Comput. 68 (2016), pp. 1281–1298.
   Web of Science ®Google Scholar
• H. Brunner, X. Wu and J. Zhang, Computational solution of blow-up problems for semilinear parabolic PDEs on unbounded domains, SIAM J. Sci. Comput. 31 (2010), pp. 4478–4496.
   Web of Science ®Google Scholar
• C. Chen, V. Thom e´e and L. B.Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comput. 58 (1992), pp. 587–602.
   Web of Science ®Google Scholar
• J. Dougias and B.F. Jones, Numerical methods for integro-differential equations of parabolic and hyperbolic types, Numerische Mathematik 4 (1962), pp. 96–102.
   Google Scholar
• H. Engler, On Some Parabolic Integro-Differential Equations: Existence and asymptotics of Solutions, in: Equadiff 82, in: Lecture Notes in Mathematics, 1017, Springer, Berlin, (1983), 161–167.
   Google Scholar
• F. Fakhar-Izadi and M. Dehghan, The spectral methods for parabolic Volterra integro-differential equations, J. Comput. Appl. Math. 235 (2011), pp. 4032–4046.
   Web of Science ®Google Scholar
• C.M. Fonseca, On the eigenvalues of some tridiagonal matrices, J. Comput. Appl. Math. 200 (2007), pp. 283–286.
   Web of Science ®Google Scholar
• C.E. Froberg, Introduction to Numerical Analysis, 2nd ed., Addison-Wesley, 1969.
   Google Scholar
• B. Fornberg and J.A. Reeger, An improved Gregory-like method for 1-D quadrature, Numerische Mathematik. 141 (2019), pp. 1–19.
   Web of Science ®Google Scholar
• H. Guo and H. Rui, Least-squares Galerkin procedures for parabolic integro-differential equations, Appl. Math. Comput. 150 (2004), pp. 749–762.
   Web of Science ®Google Scholar
• H. Guo, J. Zhang and H. Fu, Two splitting positive definite mixed finite element methods for parabolic integro-differential equations, Appl. Math. Comput. 218 (2012), pp. 11255–11268.
   Web of Science ®Google Scholar
• R.W. Hamming and R.S. Pinkhan, A class of integration formulas, Assoc. Comput. Machinery. 13 (1966), pp. 430–438.
   Google Scholar
• H. Han, L. Zhu, H. Brunner and J. Ma, The numerical solution of parabolic Volterra integro-differential equations on unbounded spatial domains, Appl. Numer. Math. 55 (2005), pp. 83–99.
   Web of Science ®Google Scholar
• H. Han, L. Zhu, H. Brunner and J. Ma, Artificial boundary conditions for parabolic Volterra integro-differential equations on unbounded two-dimensional domains, J. Comput. Appl. Math. 197 (2006), pp. 406–420.
   Web of Science ®Google Scholar
• J.D. Lambert, Computational methods in ordinary differential equations, Numerische Mathematik, 1973.
   Web of Science ®Google Scholar
• W. Li and X. Da, Finite central difference/Finite element approximations for parabolic integro-differential equations, Comput. 90 (2010), pp. 89–111.
   Google Scholar
• J. Ma, Finite element methods for partial Volterra integro-differential equations on two-dimensional unbounded spatial domains, Appl. Math. Comput. 186 (2007), pp. 598–609.
   Web of Science ®Google Scholar
• J. Matthys, A-Stable linear multistep methods for Volterra integro-differential equations, Numerische Mathematik. 27 (1976), pp. 85–94.
   Web of Science ®Google Scholar
• W. Mclean, H. Sloan and V. Thom e´e, Time discretization via laplace transformation of an integro-differential equation of parabolic type, Numerische Mathematik. 102 (2006), pp. 497–522.
   Web of Science ®Google Scholar
• K. Mustapha, H. Brunner, H. Mustapha and D. Schotzanu, An hp-Version discontinuous galerkin method for integro-differential equations of parabolic type, SIAM J. Numer. Anal. 49 (2011), pp. 1369–1396.
   Web of Science ®Google Scholar
• G.M. Phillips, Gregorys method for numerical integration, Am. Math. Soc. 79 (1972), pp. 270–274.
   Google Scholar
• A.K. Pani, V. Thom e´e and L.B. Wahlbin, Numerical methods for hyperbolic and parabolic integro-differetial equations, J. Integral Equ. Appl. 4 (1992), pp. 533–584.
   Google Scholar
• A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, 2nd ed., 1978.E3..0 KHG0dition
   Google Scholar
• G.M.M. Reddy, R.K. Sinha and J.A. Cuminato, A posteriori error analysis of the crank-nicolson finite element method for parabolic integro-differential equations, J. Sci. Comput. 79 (2019), pp. 414–441.
   Web of Science ®Google Scholar
• D. Rostamy, The new streamline diffusion for the 3D coupled schrodinger equations with a cross-phase modulation, Anziam J. 55 (2013), pp. 51–78.
   Google Scholar
• D. Rostamy, M. Jabbari and M. Gadirian, Worst case error for integro-differential equations by a lattice-nystr o¨m method, Monte Carlo Methods Appl. 19 (2013), pp. 281–330.
   Google Scholar
• D. Rostamy and S. Qasemi, Discontinuous petrov-galerkin and Bernstein-Legendre polynomials method for solving fractional damped heat- and wave-like equations, Transp. Theory Stat. Phys. 44 (2015), pp. 1–23.
   Google Scholar
• J. Steinberg, Numerical solution of Volterra integral equation, Numerische Mathematik. 19 (1972), pp. 212–217.
   Web of Science ®Google Scholar
• T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math. 11 (1993), pp. 309–319.
   Web of Science ®Google Scholar
• P.J. Vander Houwen and H.J.J. te Riele, Backward differentiation type formulas for Volterra integral equations of the second kind, Numerische Mathematik. 37 (1981), pp. 205–217.
   Web of Science ®Google Scholar
• S.D. Yang and W.H. Hong, An H1-Galerkin nonconforming mixed finite element method for integro-differential equation of parabolic type, J. Math. Res. Exposition Sept. 29 (2009), pp. 871–881.
   Google Scholar
• J.W. Zhang, H.D. Han and H. Brunner, Numerical blow-Up of semilinear parabolic PDEs on unbounded domains in R2, J. Sci. Comput. 49 (2011), pp. 367–382.
   Web of Science ®Google Scholar
• Wu Zhuoqun, Yin Jingxue, Li Huilai and Junning Zhao, Nonlinear Diffusion Equations, World Scientific, November 2000.
   Google Scholar