## THERMAL SCIENCE

International Scientific Journal

### A MODIFIED EXP-FUNCTION METHOD FOR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

**ABSTRACT**

This paper proposes a novel exponential rational function method, a modification of the well-known exp-function method, to find exact solutions of the time fractional Cahn-Allen equation and the time fractional Phi-4 equation. The solution procedure is reduced to solve a system of algebraic equations, which is then solved by Wu’s method. The results show that the present method is effective, and can be applied to other fractional differential equations.

**KEYWORDS**

PAPER SUBMITTED: 2020-04-28

PAPER REVISED: 2020-06-18

PAPER ACCEPTED: 2020-06-18

PUBLISHED ONLINE: 2021-01-31

**THERMAL SCIENCE** YEAR

**2021**, VOLUME

**25**, ISSUE

**Issue 2**, PAGES [1237 - 1241]

- Tian, A. H., et al., Fractional Prediction of Ground Temperature Based on Soil Field Spectrum, Thermal Science, 24 (2020), 4, pp. 2301-2309
- Wang, K. L., Yao, S. W., He's Fractional Derivative for the Evolution Equation, Thermal Science, 24 (2020), 4, pp. 2507-2513
- Shen, Y., El-Dib, Y. O., A Periodic Solution of the Fractional Sine-Gordon Equation Arising in Architectural Engineering, Journal of Low Frequency Noise Vibration and Active Control, On-line first, doi.org/10.1177/1461348420917565, 2020
- He, J. H., The Simpler, the Better: Analytical Methods for Non-Linear Oscillators and Fractional Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp.1252-1260
- He, J. H., Exp-Function Method for Fractional Differential Equations, International Journal of Non-Linear Sciences and Numerical Simulation, 14 (2013), 6, pp. 363-366
- Ji, F.Y., et al., A Fractal Boussinesq Equation for Non-Linear Transverse Vibration of a Nanofiber-Reinforced Concrete Pillar, Applied Mathematical Modelling, 82 (2020), June, pp. 437-448
- He, J. H., et al., Difference Equation vs. Differential Equation on Different Scales, International Journal of Numerical Methods for Heat and Fluid-Flow, On-line first, doi.org/10.1108/HFF-03-2020-0178, 2020
- Zhang, S., et al., Simplest Exp-Function Method for Exact Solutions of Mikhauilov-Novikov-Wang Equation, Thermal Science, 23 (2019), 4, pp. 2381-2388
- Yang, Y.-J., The Fractional Residual Method for Solving the Local Fractional Differential Equations, Thermal Science, 24 (2020), 4, pp. 2535-2542
- He, J. H., Homotopy Perturbation Method: A New Non-Linear Analytical Technique, Applied Mathematics and Computation, 135 (2003), 1, pp.73-79
- Yu, D. N., et al., Homotopy Perturbation Method with an Auxiliary Parameter for Non-Linear Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1540-1554
- Kuang, W. X., et al., Homotopy Perturbation Method with an Auxiliary Term for the Optimal Design of a Tangent Non-Linear Packaging System, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1075-1080
- He, J. H., Latifizadeh, H., A General Numerical Algorithm for Non-Linear Differential Equations by the Variational Iteration Method, International Journal of Numerical Methods for Heat and Fluid-Flow, 30 (2020), 11, pp. 4797-4810
- Anjum, N., He, J. H. Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), June, pp. 134-138
- Yang, Y.-J., The Local Fractional Variational Iteration Method a Promising Technology for Fractional Calculus, Thermal Science, 24 (2020), 4, pp. 2605-2614
- He, J. H., Notes on the Optimal Variational Iteration Method, Applied Mathematics Letters, 25 (2012), 10, pp.1579-1581
- He, J. H., Jin, X., A Short Review on Analytical Methods for the Capillary Oscillator in a Nanoscale Deformable Tube, Mathematical Methods in the Applied Sciences, On-line first, doi.org/10.1002/mma.6321, 2020
- He, J. H. A Short rRview on Analytical Methods For to a Fully Fourth Order Non-Linear Integral Boundary Value Problem with Fractal Derivatives, International Journal of Numerical Methods for Heat and Fluid-Flow, 30 (2020), 11, pp. 4933-4934
- Tariq, H., Akram, G ., New Approach for Exact Solutions of Time Fractional Cahn-Allen Equation and Time Fractional Phi-4 Equation, Physics A, 473 (2017), May, pp. 352-362
- Hosseini, K., et al., New Exact Travelling Wave Solutions of the Tzitzeica Type Equations Using a Novel Exponential Rational Function Method, Optic, 148 (2017), Nov., pp. 85-89
- Wu, W. T., Mathematics Mechanization, Science Press, Beijing, China, 2000
- He, J. H., et al., Geometrical Explanation of the Fractional Complex Transform and Derivative Chain Rule for Fractional Calculus, Physics Letters A, 376 (2012), 4, pp. 257-259
- He, J. H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equation, Thermal Science, 16 (2012), 2, pp. 331-334
- Li, Z. B., et al., Exact Solution of Time-Fractional Heat Conduction Equation by the Fractional Complex Transform, Thermal Science, 16 (2012), 2, pp. 335-338
- He, J. H., Ain, Q. T., New Promises and Future Challenges of Fractal Calculus: From Two-Scale Thermodynamics to Fractal Variational Principle, Thermal Science, 24 (2020), 2A, pp. 659-681
- He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
- Ain, Q. T, He, J. H., On Two-Scale Dimension and Its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
- Wang, K. L., Yao, S. W., Numerical Method for Fractional Zakharov-Kuznetsov Equations with He's Fractional Derivative, Thermal Science, 23 (2019), 4, pp. 2163-2170
- He, J. H., et al., A New Fractional Derivative and Its Application Explanation of Polar Bear Hairs, Journal of King Saud University Science, 28 (2016), 2, pp. 190-192
- He, J. H., Li, Z. B., A Fractional Model for Dye Removal, Journal of King Saud University Science, 28 (2016), 1, pp. 14-16
- He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- Wang, Q. L., et al., Fractal Calculus and Its Application Explanation of Biomechanism of Polar Hairs (Vol. 26, 1850086, 2018), Fractals, 27 (2019), 5, 1992001
- Wang, Q. L., et al., Fractal Calculus and Its aApplication Explanation of Biomechanism of Polar Hairs (Vol. 26, 1850086, 2018), Fractals, 26 (2018), 6, 1850086
- He, J. H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
- Tian, Y., Wang, K. L., Polynomial Characteristic Method an Easy Approach to Lie Symmetry, Thermal Science, 24 (2020), 4, pp. 2629-2635
- Tian, Y., Wang, K.-L., Conservation Laws for Partial Differential Equations Based on the Polynomial Characteristic Method, Thermal Science, 24 (2020), 4, pp. 2529-2534
- Zhu, L., The Quenching Behavior for a Quasilinear Parabolic Equation with Singular Source and Boundary Flux, Journal of Dynamical and Control Systems, 25 (2019), 4, pp. 519-526
- Zhu, L., Complete Quenching Phenomenon for a Parabolic p-Laplacian Equation with a Weighted Absorption, Journal of Inequalities and Applications, 2018 (2018), 1, 248