Inscrição na biblioteca: Guest
Portal Digital Begell Biblioteca digital da Begell eBooks Diários Referências e Anais Coleções de pesquisa
Journal of Porous Media
Fator do impacto: 1.49 FI de cinco anos: 1.159 SJR: 0.43 SNIP: 0.671 CiteScore™: 1.58

ISSN Imprimir: 1091-028X
ISSN On-line: 1934-0508

Volume 23, 2020 Volume 22, 2019 Volume 21, 2018 Volume 20, 2017 Volume 19, 2016 Volume 18, 2015 Volume 17, 2014 Volume 16, 2013 Volume 15, 2012 Volume 14, 2011 Volume 13, 2010 Volume 12, 2009 Volume 11, 2008 Volume 10, 2007 Volume 9, 2006 Volume 8, 2005 Volume 7, 2004 Volume 6, 2003 Volume 5, 2002 Volume 4, 2001 Volume 3, 2000 Volume 2, 1999 Volume 1, 1998

Journal of Porous Media

DOI: 10.1615/JPorMedia.2019028970
pages 411-434


Omar Abu Arqub
Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan
Nabil Shawagfeh
Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan


Time-fractional partial differential equations describe different phenomena in statistical physics, applied mathematics, and engineering. In this article, we propose and analyze an efficient reproducing kernel algorithm for the numerical solutions of such equations in porous media with Dirichlet boundary conditions. The representation of the exact and the numerical solutions is given in the W (Ω) and H (Ω) inner product spaces, whereas the computation of the required grid points relies on the R(y;s) (x; t) and r(y;s) (x; t) reproducing kernel functions. To confirm the accuracy of the new approach, several numerical examples are implemented, including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. The convergence of the utilized algorithm is studied theoretically and numerically by comparing the exact solutions of the problems under investigation. Finally, the obtained results show the significant improvement of the algorithm while saving the convergence accuracy and time.


  1. Abu Arqub, O., The Reproducing Kernel Algorithm for Handling Differential Algebraic Systems of Ordinary Differential Equations, Math. Methods Appl. Sci., vol. 39, pp. 4549–4562, 2016a.

  2. Abu Arqub, O., Approximate Solutions of DASs with Nonclassical Boundary Conditions using Novel Reproducing Kernel Algorithm, Fundamenta Informaticae, vol. 146, pp. 231–254, 2016b.

  3. Abu Arqub, O., Fitted Reproducing Kernel Hilbert Space Method for the Solutions of Some Certain Classes of Time-Fractional Partial Differential Equations Subject to Initial and Neumann Boundary Conditions, Comput. Math. Appl., vol. 73, pp. 1243– 1261, 2017a.

  4. Abu Arqub, O., Adaptation of Reproducing Kernel Algorithm for Solving Fuzzy Fredholm-Volterra Integrodifferential Equations, Neural Comput. Appl., vol. 28, pp. 1591–1610, 2017b.

  5. Abu Arqub, O., Numerical Solutions for the Robin Time-Fractional Partial Differential Equations of Heat and Fluid Flows based on the Reproducing Kernel Algorithm, Int. J. Numer. Methods Heat Fluid Flow, vol. 28, pp. 828–856, 2018.

  6. Abu Arqub, O. and Al-Smadi, M., Numerical Algorithm for Solving Two-Point, Second-Order Periodic Boundary Value Problems for Mixed Integro-Differential Equations, Appl. Math. Comput., vol. 243, pp. 911–922, 2014.

  7. Abu Arqub, O., Al-Smadi, M., Momani, S., and Hayat, T., Numerical Solutions of Fuzzy Differential Equations using Reproducing Kernel Hilbert Space Method, Soft Comput., vol. 20, pp. 3283–3302, 2016.

  8. Abu Arqub, O., Al-Smadi, M., Momani, S., and Hayat, T., Application of Reproducing Kernel Algorithm for Solving Second- Order, Two-Point Fuzzy Boundary Value Problems, Soft Comput., vol. 21, pp. 7191–7206, 2017.

  9. Abu Arqub, O., Al-Smadi, M., and Shawagfeh, N., Solving Fredholm Integro-Differential Equations using Reproducing Kernel Hilbert Space Method, Appl. Math. Comput., vol. 219, pp. 8938–8948, 2013.

  10. Abu Arqub, O., El-Ajou, A., and Momani, S., Constructing and Predicting Solitary Pattern Solutions for Nonlinear Time-Fractional Dispersive Partial Differential Equations, J. Comput. Phys., vol. 293, pp. 385–399, 2015.

  11. Abu Arqub, O. and Maayah, B., Solutions of Bagley-Torvik and Painlev´e Equations of Fractional Order using Iterative Reproducing Kernel Algorithm, Neural Comput. Appl., vol. 29, pp. 1465–1479, 2018a.

  12. Abu Arqub, O. and Maayah, B., Numerical Solutions of Integrodifferential Equations of Fredholm Operator Type in the Sense of the Atangana–Baleanu Fractional Operator, Chaos, Solitons Fractals, vol. 117, pp. 117–124, 2018b.

  13. Abu Arqub, O. and Rashaideh, H., The RKHS Method for Numerical Treatment for Integrodifferential Algebraic Systems of Temporal Two-Point BVPs, Neural Comput. Appl., vol. 30, pp. 2595–2606, 2018.

  14. Al-Smadi, M., Abu Arqub, O., Shawagfeh, N., and Momani, S., Numerical Investigations for Systems of Second-Order Periodic Boundary Value Problems using Reproducing Kernel Method, Appl. Math. Comput., vol. 291, pp. 137–148, 2016.

  15. Aronszajn, N., Theory of Reproducing Kernels, Transact. Am. Math. Soc., vol. 68, pp. 337–404, 1950.

  16. Berlinet, A. and Agnan, C.T., Reproducing Kernel Hilbert Space in Probability and Statistics, Boston: Kluwer Academic Publishers, 2004.

  17. Chen, A. and Li, C., Numerical Solution of Fractional Diffusion-Wave Equation, Numer. Funct. Anal. Optim., vol. 37, pp. 19–39, 2016.

  18. Chen, J., Liu, F., and Anh, V., Analytical Solution for the Time-Fractional Telegraph Equation by the Method of Separating Variables, J. Math. Anal. Appl., vol. 338, pp. 1364–1377, 2008.

  19. Comptedag, A. and Metzlerddag, R., The Generalized Cattaneo Equation for the Description of Anomalous Transport Processes, J. Physics A: Math. General, vol. 30, pp. 7277–7289, 1997.

  20. Cui, M. and Lin, Y., Nonlinear Numerical Analysis in the Reproducing Kernel Space, New York: Nova Science, 2009.

  21. Daniel, A., Reproducing Kernel Spaces and Applications, Basel, Switzerland: Springer, 2003.

  22. Das, S., Functional Fractional Calculus for System Identification and Controls, Berlin: Springer, 2008.

  23. Dehghan, M. and Safarpoor, M., The Dual Reciprocity Boundary Integral Equation Technique to Solve a Class of the Linear and Nonlinear Fractional Partial Differential Equations, Math. Methods Appl. Sci., vol. 39, pp. 2461–2476, 2016.

  24. Diethelm, K., The Analysis of Fractional Differential Equations, Berlin: Springer, 2010.

  25. El-Ajou, A., Abu Arqub, O., and Momani, S., Approximate Analytical Solution of the Nonlinear Fractional KdV-Burgers Equation: A New Iterative Algorithm, J. Comput. Phys., vol. 293, pp. 81–95, 2015a.

  26. El-Ajou, A., Abu Arqub, O., Momani, S., Baleanu, D., and Alsaedi, A., A Novel Expansion Iterative Method for Solving Linear Partial Differential Equations of Fractional Order, Appl. Math. Comput., vol. 257, pp. 119–133, 2015b.

  27. Geng, F.Z. and Cui, M., A Reproducing Kernel Method for Solving Nonlocal Fractional Boundary Value Problems, Appl. Math. Lett., vol. 25, pp. 818–823, 2012.

  28. Geng, F.Z. and Qian, S.P., Reproducing Kernel Method for Singularly Perturbed Turning Point Problems Having Twin Boundary Layers, Appl. Math. Lett., vol. 26, pp. 998–1004, 2013.

  29. Geng, F.Z. and Qian, S.P., Modified Reproducing Kernel Method for Singularly Perturbed Boundary Value Problems with a Delay, Appl. Math. Modell., vol. 39, pp. 5592–5597, 2015.

  30. Geng, F.Z., Qian, S.P., and Li, S., A Numerical Method for Singularly Perturbed Turning Point Problems with an Interior Layer, J. Comput. Appl. Math., vol. 255, pp. 97–105, 2014.

  31. Golmankhaneh, A.K., Golmankhaneh, A.K., and Baleanu, D., On Nonlinear Fractional Klein–Gordon Equation, Signal Process., vol. 91, pp. 446–451, 2011.

  32. Hilfer, R., Applications of Fractional Calculus in Physics, Singapore: World Scientific, 2000.

  33. Jiang, W. and Chen, Z., Solving a System of Linear Volterra Integral Equations using the New Reproducing Kernel Method, Appl. Math. Computat., vol. 219, pp. 10225–10230, 2013.

  34. Jiang, W. and Chen, Z., A Collocation Method based on Reproducing Kernel for a Modified Anomalous Subdiffusion Equation, Numer. Methods Partial Diff. Eqs., vol. 30, pp. 289–300, 2014.

  35. Jiang,W. and Lin, Y., Approximate Solution of the Fractional Advection-Dispersion Equation, Comput. Phys. Commun., vol. 181, pp. 557–561, 2010.

  36. Jiang, W. and Lin, Y., Representation of Exact Solution for the Time Fractional Telegraph Equation in the Reproducing Kernel Space, Commun. Nonlinear Sci. Numer. Simul., vol. 16, pp. 3639–3645, 2011.

  37. Kumar, S., Kumar, D., and Singh, J., Numerical Computation of Fractional Black-Scholes Equation Arising in Financial Market, Egypt. J. Basic Appl. Sci., vol. 1, pp. 177–183, 2014.

  38. Kumar, S. and Rashidi, M.M., New Analytical Method for Gas Dynamics Equation Arising in Shock Fronts, Comput. Phys. Commun., vol. 185, pp. 1947–1954, 2014.

  39. Kumar, D., Singh, J., and Baleanu, D., Numerical Computation of a Fractional Model of Differential-Difference Equation, J. Comput. Nonlinear Dynam., vol. 11, p. 061004, 2016.

  40. Kumar, D., Singh, J., Kumar, S., Sushila, and Singh, B.P., Numerical Computation of Nonlinear ShockWave Equation of Fractional Order, Ain Shams Eng. J., vol. 6, pp. 605–611, 2015.

  41. Kilbas, A.A.A., Srivastava, H.M., and Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.

  42. Lin, Y., Cui, M., and Yang, L., Representation of the Exact Solution for a Kind of Nonlinear Partial Differential Equations, Appl. Math. Lett., vol. 19, pp. 808–813, 2006.

  43. Li, C. and Deng, W., Chaos Synchronization of Fractional-Order Differential Systems, Int. J. Modern Phys. B, vol. 20, pp. 791– 803, 2006.

  44. Li, X. and Wu, B., Approximate Analytical Solutions of Nonlocal Fractional Boundary Value Problems, Appl. Math. Modell., vol. 39, pp. 1717–1724, 2015.

  45. Magin, R.L., Fractional Calculus Models of Complex Dynamics in Biological Tissues, Comput. Math. Appl., vol. 59, pp. 1586– 1593, 2010.

  46. Metzler, R. and Klafter, J., The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach, Phys. Rep., vol. 339, pp. 1–77, 2000.

  47. Mohebbi, A. and Dehghan, M., High-Order Solution of One-Dimensional Sine-Gordon Equation using Compact Finite Difference and DIRKN Methods, Math. Comput. Modell., vol. 51, pp. 537–549, 2010.

  48. Momani, S., Abu Arqub, O., Hayat, T., and Al-Sulami, H., A Computational Method for Solving Periodic Boundary Value Problems for Integro-Differential Equations of Fredholm–Volterra Type, Appl. Math. Comput., vol. 240, pp. 229–239, 2014.

  49. Odibat, Z. and Momani, S., The Variational Iteration Method: An Efficient Scheme for Handling Fractional Partial Differential Equations in Fluid Mechanics, Comput. Math. Appl., vol. 58, pp. 2199–2208, 2009.

  50. Ortigueira, M.D. and Machado, J.A.T., Fractional Signal Processing and Applications, Signal Process., vol. 83, pp. 2285–2286, 2003.

  51. Rehman, M.U. and Khan, R.A., Numerical Solutions to Initial and Boundary Value Problems for Linear Fractional Partial Differential Equations, Appl. Math. Modell., vol. 37, pp. 5233–5244, 2013.

  52. Singh, J., Kumar, D., and Nieto, J.J., A Reliable Algorithm for Local Fractional Tricomi Equation Arising in Fractal Transonic Flow, Entropy, vol. 18, no. 6, p. 206, 2016a. DOI: 10.3390/e18060206

  53. Singh, J., Kumar, D., and Swroop, R., Numerical Solution of Time- and Space-Fractional Coupled Burgers Equations via Homotopy Algorithm, Alexandria Eng. J., vol. 55, pp. 1753–1763, 2016b.

  54. Singh, J., Rashidi, M.M., Kumar, D., and Swroop, R., A Fractional Model of a Dynamical Brusselator Reaction-Diffusion System Arising in Triple Collision and Enzymatic Reactions, Nonlinear Engineer. Model. Appl., vol. 5, pp. 277–285, 2016c.

  55. Srivastava, H.M., Kumar, D., and Singh, J., An Efficient Analytical Technique for Fractional Model of Vibration Equation, Appl. Math. Modell., vol. 45, pp. 192–204, 2017.

  56. Sun, Z.Z. and Wu, X., A Fully Discrete Difference Scheme for a Diffusion-Wave System, Appl. Numer. Math., vol. 56, pp. 193– 209, 2006.

  57. Tarasova, V.E. and Zaslavsky, G.M., Fractional Dynamics of Systems with Long-Range Space Interaction and Tempolar Memory, Physica A: Statist. Mech. Appl., vol. 383, pp. 291–308, 2007.

  58. Vong, S. and Wang, Z., A High-Order Compact Scheme for the Nonlinear Fractional Klein–Gordon Equation, Numer. Methods Part. Diff. Eqs., vol. 31, pp. 706–722, 2015.

  59. Weinert, H.L., Reproducing Kernel Hilbert Spaces: Applications in Statistical Signal Processing, London: Hutchinson, 1982.

  60. Wang, Q.F. and Cheng, D., Numerical Solution of Damped Nonlinear Klein-Gordon Equations using Variational Method and Finite Element Approach, Appl. Math. Comput., vol. 162, pp. 381–401, 2005.

  61. Wang, Y., Du, M., Tan, F., Li, Z., and Nie, T., Using Reproducing Kernel for Solving a Class of Fractional Partial Differential Equation with Non-Classical Conditions, Appl. Math. Comput., vol. 219, pp. 5918–5925, 2013a.

  62. Wang, W., Han, B., and Yamamoto, M., Inverse Heat Problem of Determining Time-Dependent Source Parameter in Reproducing Kernel Space, Nonlinear Anal.: Real World Appl., vol. 14, pp. 875–887, 2013b.

  63. Yang, L.H. and Lin, Y., Reproducing Kernel Methods for Solving Linear Initial-Boundary-Value Problems, Electron. J. Diff. Eqs., vol. 2008, pp. 1–11, 2008.

  64. Yang, L.H., Shen, J.H., and Wang, Y., The Reproducing Kernel Method for Solving the System of the Linear Volterra Integral Equations with Variable Coefficients, J. Comput. Appl. Math., vol. 236, pp. 2398–2405, 2012.

  65. Yusufoglu, E., The Variational Iteration Method for Studying the Klein-Gordon Equation, Appl. Math. Lett., vol. 4, pp. 669–674, 2008.

  66. Zhoua, Y., Cui, M., and Lin, Y., Numerical Algorithm for Parabolic Problems with Non-Classical Conditions, J. Comput. Appl. Math., vol. 230, pp. 770–780, 2009.