## THERMAL SCIENCE

International Scientific Journal

## Authors of this Paper

,

### NUMERICAL INVERSE LAPLACE HOMOTOPY TECHNIQUE FOR FRACTIONAL HEAT EQUATIONS

ABSTRACT
In this paper, we have aimed the numerical inverse Laplace homotopy technique for solving some interesting 1-D time-fractional heat equations. This method is based on the Laplace homotopy perturbation method, which is combined form of the Laplace transform and the homotopy perturbation method. Firstly, we have applied to the fractional 1-D PDE by using He’s polynomials. Then we have used Laplace transform method and discussed how to solve these PDE by using Laplace homotopy perturbation method. We have declared that the proposed model is very efficient and powerful technique in finding approximate solutions to the fractional PDE.
KEYWORDS
PAPER SUBMITTED: 2017-08-04
PAPER REVISED: 2017-11-15
PAPER ACCEPTED: 2017-11-20
PUBLISHED ONLINE: 2018-01-07
DOI REFERENCE: https://doi.org/10.2298/TSCI170804285Y
THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Supplement 1, PAGES [S185 - S194]
REFERENCES
1. Kumar, D., et. al., A New Analysis for Fractional Model of Regularized Long‐Wave Equation Arising in Ion Acoustic Plasma Waves, Mathematical Methods in the Applied Sciences, (2017),
2. Inc, M., He's Homotopy Perturbation Method for Solving Korteweg‐De Vries Burgers Equation with Initial Condition, Numerical Methods for Partial Differential Equations, 26 (2010), 5, pp. 1224-1235
3. Özdemir, N. and Yavuz, M., Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation, Acta Physica Polonica A, 132 (2017), 3, pp. 1050-1053
4. Yavuz, M., et. al., Generalized Differential Transform Method for Fractional Partial Differential Equation from Finance, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, 2016, pp. 778-785
5. Yerlikaya-Özkurt, F., et. al., Estimation of the Hurst Parameter for Fractional Brownian Motion Using the Cmars Method, Journal of Computational and Applied Mathematics, 259 (2014), 843-850
6. Kumar, S., et. al., Two Analytical Methods for Time-Fractional Nonlinear Coupled Boussinesq-Burger's Equations Arise in Propagation of Shallow Water Waves, Nonlinear Dynamics, 85 (2016), 2, pp. 699-715
7. Ibrahim, R. W., On Holomorphic Solutions for Nonlinear Singular Fractional Differential Equations, Computers & Mathematics with Applications, 62 (2011), 3, pp. 1084-1090
8. Özdemir, N., et. al., The Numerical Solutions of a Two-Dimensional Space-Time Riesz- Caputo Fractional Diffusion Equation, An International Journal of Optimization and Control, 1 (2011), 1, pp. 17-26
9. Eroğlu, B. İ., et. al., Optimal Control Problem for a Conformable Fractional Heat Conduction Equation, Acta Physica Polonica A, 132 (2017), 3, pp. 658-662
10. Evirgen, F., Conformable Fractional Gradient Based Dynamic System for Constrained Optimization Problem, Acta Physica Polonica A, 132 (2017), 3, pp. 1066-1069
11. Hu, Y., et. al., Optimal Consumption and Portfolio in a Black-Scholes Market Driven by Fractional Brownian Motion, Infinite dimensional analysis, quantum probability and related topics, 6 (2003), 04, pp. 519-536
12. Özdemir, N., et. al., Analysis of an Axis-Symmetric Fractional Diffusion-Wave Problem, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 35, pp. 355208
13. Jordan, H., Steady-State Heat Conduction in a Medium with Spatial Non-Singular Fading Memory: Derivation of Caputo-Fabrizio Space-Fractional Derivative from Cattaneo Concept with Jeffrey's Kernel and Analytical Solutions, Thermal Science, 21 (2017), 2, pp. 827-839
14. Avci, D., et. al., Conformable Heat Equation on a Radial Symmetric Plate, Thermal Science, 21 (2017), 2, pp. 819-826
15. Inc, M. and Cavlak, E., He's Homotopy Perturbation Method for Solving Coupled-Kdv Equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10 (2009), 3, pp. 333-340
16. Morales-Delgado, V. F., et. al., Laplace Homotopy Analysis Method for Solving Linear Partial Differential Equations Using a Fractional Derivative with and without Kernel Singular, Advances in Difference Equations, 2016 (2016), 1, pp. 164
17. Inc, M. and Uğurlu, Y., Numerical Simulation of the Regularized Long Wave Equation by He's Homotopy Perturbation Method, Physics Letters A, 369 (2007), 3, pp. 173-179
18. Yavuz, M., Novel Solution Methods for Initial Boundary Value Problems of Fractional Order with Conformable Differentiation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8 (2017), 1, pp. 1-7
19. Javidi, M. and Ahmad, B., Numerical Solution of Fractional Partial Differential Equations by Numerical Laplace Inversion Technique, Advances in Difference Equations, 2013 (2013), 1, pp. 375
20. Madani, M., et. al., On the Coupling of the Homotopy Perturbation Method and Laplace Transformation, Mathematical and Computer Modelling, 53 (2011), 9, pp. 1937-1945
21. Talbot, A., The Accurate Numerical Inversion of Laplace Transforms, IMA Journal of Applied Mathematics, 23 (1979), 1, pp. 97-120
22. Povstenko, Y., et. al., Control of Thermal Stresses in Axissymmetric Problems of Fractional Thermoelasticity for an Infinite Cylindrical Domain, Thermal Science, 21 (2017), 1, pp. 19- 28
23. Evirgen, F. and Özdemir, N., A Fractional Order Dynamical Trajectory Approach for Optimization Problem with Hpm, in: Fractional Dynamics and Control (Ed. D. Baleanu, J.A.T. Machado, and A.C.J. Luo), Springer, 2012, pp. 145-155
24. Yan, L.-M., Modified Homotopy Perturbation Method Coupled with Laplace Transform for Fractional Heat Transfer and Porous Media Equations, Thermal Science, 17 (2013), 5, pp. 1409-1414
25. Zhang, M.-F., et. al., Efficient Homotopy Perturbation Method for Fractional Non-Linear Equations Using Sumudu Transform, Thermal Science, 19 (2015), 4, pp. 1167-1171
26. Torabi, M., et. al., Assessment of Homotopy Perturbation Method in Nonlinear Convective- Radiative Non-Fourier Conduction Heat Transfer Equation with Variable Coefficient, Thermal Science, 15 (2011), suppl. 2, pp. 263-274
27. Hetmaniok, E., et. al., Solution of the Inverse Heat Conduction Problem with Neumann Boundary Condition by Using the Homotopy Perturbation Method, Thermal Science, 17 (2013), 3, pp. 643-650
28. Abou-Zeid, M., Homotopy Perturbation Method to Mhd Non-Newtonian Nanofluid Flow through a Porous Medium in Eccentric Annuli with Peristalsis, Thermal Science, (2015), 00, pp. 79-79
29. Stehfest, H., Algorithm 368: Numerical Inversion of Laplace Transforms
30. He, J.-H., Homotopy Perturbation Technique, Computer methods in applied mechanics and engineering, 178 (1999), 3, pp. 257-262
31. He, J.-H., Homotopy Perturbation Method for Solving Boundary Problems, Phys. Lett. A., 350 (2006), 1-2, pp. 87-88
32. Rajabi, A., et. al., Application of Homotopy Perturbation Method in Nonlinear Heat Conduction and Convection Equations, Physics Letters A, 360 (2007), 4, pp. 570-573
33. He, J.-H., Application of Homotopy Perturbation Method to Nonlinear Wave Equations, Chaos, Solitons & Fractals, 26 (2005), 3, pp. 695-700
34. Odibat, Z. and Momani, S., The Variational Iteration Method: An Efficient Scheme for Handling Fractional Partial Differential Equations in Fluid Mechanics, Computers & Mathematics with Applications, 58 (2009), 11, pp. 2199-2208
35. Chen, W., et. al., Fractional Diffusion Equations by the Kansa Method, Computers & Mathematics with Applications, 59 (2010), 5, pp. 1614-1620