THERMAL SCIENCE

International Scientific Journal

Muhammad Aslam Noor

,

Muhammad Rafiq

,

Salah-Ud-Din Khan

,

Muhammad Amer Qureshi

,

Muhammad Kamran

,

Shahab-Ud-Din Khan

,

Faisal Saeed

,

Hijaz Ahmad

ANALYTICAL SOLUTIONS TO CONTACT PROBLEM WITH FRACTIONAL DERIVATIVES IN THE SENSE OF CAPUTO

ABSTRACT
The current study extends the applications of the variational iteration method for the analytical solution of fractional contact problems. The problem involves Caputo sense while calculating the derivative of fractional order, we apply the Penalty function technique to transform it into a system of fractional boundary value problems coupled with a known obstacle. The variational iteration method is employed to find the series solution of fractional boundary value problem. For different values of fractional parameters, residual errors of solutions are plotted to make sure the convergence and accuracy of the solution. The reasonably accurate results show that one of the highly effective and stable methods for the solution of fractional boundary value problem is the method of variational iteration.
KEYWORDS
contact problem, obstacle, variational iteration method, fractional derivative
PAPER SUBMITTED: 2020-04-25
PAPER REVISED: 2020-06-19
PAPER ACCEPTED: 2020-06-30
PUBLISHED ONLINE: 2020-10-25
DOI REFERENCE: https://doi.org/10.2298/TSCI20S1313N
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE Supplement 1, PAGES [S313 - S323]
REFERENCES
  1. Podlubny, I., Fractional Differential Equations: An Introduction, in: Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, New York, USA, 1998
  2. Freihet, A., et al., Construction of Fractional Power Series Solutions to Fractional Stiff System Using Residual Functions Algorithm, Adv. Differ Equations, 2019 (2019), 1, 95
  3. Wang, Q., Numerical Solutions for Fractional kdv-Burgers Equation by Adomian Decomposition Method, Appl. Math. Comput., 182 (2006), 2, pp. 1048-1055
  4. Zurigat, M., et al., Analytical Approximate Solutions of Systems of Fractional Algebraic-Differential Equations by Homotopy Analysis Method, Comput. Math. with Appl., 59 (2010), 3, pp. 1227-1235
  5. Iomin, A., Fractional Evolution in Quantum Mechanics. Chaos, Solitons Fractals, X (2019), 1, 100001
  6. Turut, V., Guzel, N., Multivariate Pade Approximation for Solving Non-Linear Partial Differential Equations of Fractional Order, In Abstr. Appl. Anal., 2013 (2013), ID746401
  7. Liu, J., Hou, G., Numerical Solutions of the Space-and Time-Fractional Coupled Burgers Equations by Generalized Differential Transform Method, Appl. Math. Comput., 217 (2011), 16, pp. 7001-7008
  8. Abbas, M., et al., The Application of Cubic Trigonometric b-Spline to the Numerical Solution of the Hyperbolic Problems, Appl Math. Comput., 239 (2014), July, pp. 74-88
  9. Inc, M., et al., Modified Variational Iteration Method for Straight Fins with Temperature Dependent Thermal Conductivity, Thermal Science, 22 (2018), Suppl. 1, pp. S229-S236
  10. Partohaghighi, M., et al., Ficitious Time Integration Method for Solving the Time Fractional Gas Dynamics Equation, Thermal Science, 23 (2019), Suppl. 6, pp. S2009-S2016
  11. Kilicman, A., et al., Analytic Approximate Solutions for Fluid-Flow in the Presence of Heat and Mass Transfer, Thermal Science, 22 (2018), Suppl. 1, pp. S259-S264
  12. Jassim, H. K., Baleanu, D., A Novel Approach for Korteweg-de Vries Equation of Fractional Order, Journal Appl. Comput. Mech., 5 (2019), 2, pp. 192-198
  13. Baashan, A., et al., Approximation of the kdvb Equation by the Quintic b-Spline Differential Quadrature Method, Kuwait J. Sci., 42 (2015), 2, pp. 67-92
  14. Ramadan, M., et al., Quintic Non-Polynomial Spline Solutions for Fourth Order Two-Point Boundary Value Problem, Commun Non-Linear Sci. Numer. Simul., 14 (2009), 4, pp. 1105-1114
  15. Khan, A., Sultana, T., Non-Polynomial Ouintic Spline Solution for the System of Third Order Boundary-Value Problems, Numer. Algorithms, 59 (2012), 4, pp. 541-559
  16. Srivastava, P. K., Study of Differential Equations with Their Polynomial and Non-Polynomial Spline Based Approximation, Acta Tech. Corviniensis-Bulletin Eng., 7 (2014), 3, 139
  17. Siddiqi, S. S., Arshed, S., Numerical Solution of Time-Fractional Fourth Order Partial Differential Equations, Int. J. Comput. Math., 92 (2015), 7, pp. 1496-1518
  18. Li, X., Wong, P. J., An Efficient Non-Polynomial Spline Method for Distributed Order Fractional Subdiffusion Equations, Math. Methods Appl. Sci., 41 (2018), 13, pp. 4906-4922
  19. Stampacchia, G., Formes bilinéaires coercitives Sur les ensembles convexes (in Franch), Comptes Rendus Hebdomadaires Des Seances De L' Academie Des Sciences, 258 (1964), 18, 4413
  20. Hu, Y., et al., Analytical Solution of the Linear Fractional Differential Equation by Adomian Decomposition Method, Journal Comput. Appl. Math., 215 (2008), 1, pp. 220-229
  21. Rani, A., et al., Solving System of Differential Equations of Fractional Order by Homotopy Analysis Method, Journal Sci. Arts, 17 (2017), 3, pp. 457-468
  22. Inc, M., The Approximate and Exact Solutions of the Space-and Time-Fractional Burgers Equations with Initial Conditions by Variational Iteration Method, Journal Math. Anal. Appl., 345 (2008), 1, pp. 476-484
  23. Modanli, M., Akgül, A., Numerical Solution of Fractional Telegraph Differential Equations by Theta-Method, Eur. Phys. J. Spec. Top., 226 (2017), 16-18, pp. 3693-3703
  24. Martin, O., Stability Approach to the Fractional Variational Iteration Method Used for the Dynamic Analysis of Viscoelastic Beams, Journal Comput. Appl. Math., 346 (2019), Jan., pp. 261-276
  25. Heidarkhani, S., et al., A Variational Approach to Perturbed Impulsive Fractional Differential Equations, Journal Comput. Appl. Math., 341 (2018), Oct., pp. 42-60
  26. Kubica, A., Ryszewska, K., Decay of Solutions to Parabolic-Type Problem with Distributed Order Caputo Derivative, Journal Math. Anal. Appl., 465 (2018), 1, pp. 75-99
  27. Noor, M. A., et al., Homotopy Perturbation Method for Solving a System of Third-Order Boundary Value Problems, Int. J. Phys. Sci., 6 (2011), 16, pp. 4128-4133
  28. Tuck, E., Schwartz, L., A Numerical and Asymptotic Study of Some Third-Order Ordinary Differential Equations Relevant to Draining and Coating Flows, SIAM Rev., 32 (1990), 3, pp. 453-469
  29. Ahmad, H., Variational Iteration Method with an Auxiliary Parameter for Solving Differential Equations of the Fifth Order, Non-Linear Sci. Lett. A, 9 (2018), Mar., pp. 27-35
  30. Inokuti, M., et al., General Use of the Lagrange Multiplier in Non-Linear Mathematical Physics, Var. Methods Mech. Solids, 33 (1978), 5, pp. 156-162
  31. He, J.-H., Variational Iteration Method - A Kind of Non-Linear Analytical Technique: Some Examples, Int. J. Non-Linear Mech., 34 (1999), 4, pp. 699-708
  32. Ahmad, H., et al., Variational Iteration Algorithm-i with an Auxiliary Parameter for Solving Boundary Value Problems, Earthline J. Math. Sci., 3 (2020), 2, pp. 229-247
  33. Ahmad, H., Khan, T. A., Variational Iteration Algorithm-i with an Auxiliary Parameter for Wave-Like Vibration Equations, Journal Low Freq. Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1113-1124
  34. Ahmad, H., Khan, T. A., Variational Iteration Algorithm i with an Auxiliary Parameter for the Solution of Differential Equations of Motion for Simple and Damped Mass-Spring Systems, Noise Vib World, 51 (2020), 1-2, pp. 12-20
PDF VERSION [DOWNLOAD]
Analytical solutions to contact problem with fractional derivatives in the sense of Caputo

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence