International Scientific Journal


The convection differential models play an essential role in studying different chemical process and effects of the diffusion process. This paper intends to provide optimized numerical results of such equations based on the conformable fractional derivative. Subsequently, a well-known heuristic optimization technique, differential evolution algorithm, is worked out together with the Taylor’s series expansion, to attain the optimized results. In the scheme of the Taylor optimization method (TOM), after expanding the functions with the Taylor’s series, the unknown terms of the series are then globally optimized using differential evolution. Moreover, to illustrate the applicability of TOM, some examples of linear and non-linear fractional convection diffusion equations are exemplified graphically. The obtained assessments and comparative demonstrations divulged the rapid convergence of the estimated solutions towards the exact solutions. Comprising with an effective expander and efficient optimizer, TOM reveals to be an appropriate approach to solve different fractional differential equations modeling various problems of engineering.
PAPER REVISED: 2017-11-15
PAPER ACCEPTED: 2017-11-30
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THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Supplement 1, PAGES [S243 - S252]
  1. Miller, K. S., An introduction to fractional calculus and fractional differential equations, John Wiley and Sons, New York, 1993
  2. He J., H., A New Fractal Derivation, Thermal Science, 15(2011), 1, pp. S145-S147
  3. Atangana, A., Baleanu, D., New Fractional Derivatives with Nonlocal and Non-singular Kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), pp. 763-769
  4. Caputo., M., Fabrizio., M, A New Definition of Fractional Derivative without Singular Kernel, Progress in Fractional Differentiation and Application, 1(2015), 2, pp. 73-85
  5. Khalil., R., Horani., M., A, Yousef., A., Sababheh., M., A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics, 264 (2014), pp. 65-70
  6. Hosseini., K, Bekir., A, Ansari., R, New Exact Solutions of The Conformable time-Fractional Cahn- Allen and Cahn-Hilliard Equations using the Modified Kudryashov method, Optik - International Journal for Light and Electron Optics, 132 (2017), pp. 203-209
  7. Wang, K., L., Liu., S., Y., He's Fractional Derivative for Non-linear Fractional Heat Transfer Equation, Thermal Science, 20 (2016), 3, pp. 793-796
  8. Wang., K., Liu., S., Y., A New Solution Procedure for Nonlinear Fractional Porous Media Equation Based on A New Fractional Derivative, Nonlinear Science. Letters A, 7 (2016), 4, pp. 135-140
  9. Sayevand., K, Pichaghchi., K, Analysis of Nonlinear Fractional KdV Equation Based on He's Fractional Derivative, Nonlinear Science. Letters A, 7 (2016), 3, pp. 77-85
  10. Eslami., M, Rezazadeh., H., The First Integral Method for Wu-Zhang System with Conformable time-Fractional Derivative, Calcolo 53 (2016), pp. 475-485
  11. Jarad., F., Ugurlu., E., Abdelijawad., T., Baleanu., D., On A New Class of Fractional Operators, Advances in Difference Equations (2017), ArticleID 247
  12. Goufo., E., F., D., Application of The Caputo-Fabrizio Fractional Derivative Without Singular kernel to Korteweg-de Vries-Bergers Equation, Mathmatical Modelling and Analysis 21 (2016), pp. 188-198
  13. Atangana., A, Koca., I., Chaos in A Simple Nonlinear System with Atangana-Baleanu Derivatives with Fractional Order, Chaos Solitons & Fractals. 89 (2016), pp. 447-454
  14. Wu., G., C., Baleanu., D., Luo., W., H., Lyapunov Functions for Riemann-Liouville-like Fractional Difference Equations, Applied Mathematics and Computation, 314 (2017) pp. 228-236
  15. Odibat., Z., Momani., S., A Generalized Differential Transform Method for Linear Partial, Differential Equations of Fractional Order, Applied Mathematics Letters 21 (2008), pp. 194-199
  16. Abolhasani., M., Abbasbandy., S., Allahviranloo., T, A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains, Mathematics 5 (2017), 26 doi:10.3390/math5020026
  17. Chen., Y., Wu. Y, Cui., Y., Wang., Z., Jin., D, Wavelet Method for A Class of Fractional Convection-Diffusion Equation with Variable Coefficients, Journal of Computational Science 1 (2010) pp. 146-149
  18. Abbasbandya., S., Kazem., S., Alhuthali., M., S., Alsulami., H., H., Application of the Operational Matrix of Fractional-Order Legendre Functions for Solving the Time-Fractional Convection-Diffusion Equation, Applied Mathematics and Computation 266 (2015) pp. 31-40
  19. Hariharan., G., Kannan., K., Review of Wavelet Methods for the Solution of Reaction-Diffusion Problems in Science and Engineering, Applied Mathematical Modelling 38 (2014) pp. 799-813
  20. Abdusalam., H., A., Analytic and Approximate Solutions for Nagumo Telegraph Reaction Diffusion Equation, Applied Mathematics and Computation 157 (2004) pp. 515-522
  21. Wu., G., C., Baleanu., D, Luo., W. H., Analysis of Fractional Non-Linear Diffusion Behaviors Based on Adomian polynomial, Thermal Science 21(2017) pp. 813-817
  22. Saruhan., H, Differential Evolution and Simulated Annealing Algorithms For Mechanical Systems Design, Engineering Science and Technology, 17 (2014) pp. 131-136
  23. Storn., R., Price., K, Differential Evolution: A Simple and Efficient Adaptive Scheme For Global Optimization Over Continuous Spaces, International Computer Science Institute, 12 (1995) pp. 1-16
  24. Storn., R., Price., K., Lampinen., J., A., Differential Evolution-A Practical Approach to Global Optimization, Springer-Natural Computing Series. Vanderplaats, 2005
  25. Bülbül, B., Sezer, M., Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method, Journal of Applied Mathematics, 2013 (2013) ArticleID 691614
  26. Aslan, B. B., Gurbuz, B., Sezer, M., A Taylor Matrix-Collocation Method Based on Residual Error for Solving Lane-Emden Type Differential Equations, New Trends Math. Sci. 224(2015), 2, pp. 219-224

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