## THERMAL SCIENCE

International Scientific Journal

### APPROXIMATE ANALYTICAL SOLUTION TO THE KUDRYASHOV-SINELSHCHIKOV EQUATION WITH HE'S FRACTIONAL DERIVATIVE

**ABSTRACT**

In this paper, the Adomian decomposition method was employed successfully to solve the Kudryashov-Sinelshchikov equation involving He's fractional derivatives, and an approximate analytical solution was obtained.

**KEYWORDS**

PAPER SUBMITTED: 2021-08-05

PAPER REVISED: 2022-07-15

PAPER ACCEPTED: 2022-07-19

PUBLISHED ONLINE: 2023-06-11

**THERMAL SCIENCE** YEAR

**2023**, VOLUME

**27**, ISSUE

**Issue 3**, PAGES [1795 - 1802]

- Adomian, G., A Review of the Decomposition Method in Applied Mathematics, Journal of Mathematical Analysis and Applications, 135 (1988), 2, pp. 501-544
- He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- Randruut, M., On the Kudryashov-Sinelshchikov Equation for Waves in Bubbly Liquids, Physics Letters A, 42 (2011), 375, pp. 3687-3692
- Randruut, M., Braun, M., Cnoidal Waves Governed by the Kudryashov-Sinelshchikov Equation, Physics Letters A,31-33 (2013), 377, pp. 1868-1874
- Coclite, G. M., Ruvo, L. D., Existence Results for the Kudryashov-Sinelshchikov Olver Equation, Proceedings of the Royal Society of Edinburgh Section A Mathematics, 2 (2020), 151, pp. 1-26
- Seadawy, A. R., et al., Non-linear Wave Solutions of the Kudryashov-Sinelshchikov Dynamical Equation in Mixtures Liquid-Gas Bubbles Under the Consideration of Heat Transfer and Viscosity, Journal of Taibah University for Science, 1 (2019), 13, pp. 1060-1072
- Nadjafikhah, M., Shirvani-Sh, V., Lie Symmetry Analysis of Kudryashov-Sinelshchikov Equation, Mathematical Problems in Engineering ,3 (2011), 1, pp. 505-515
- Randrueuet, M., Braun, M., On Identical Traveling-Wave Solutions of the Kudryashov Sinelshchikov and Related Equations, International Journal of Non-Linear Mechanics, 1 (2014), 5, pp. 206-211
- Gupta, A. K., Ray, S. S., On the Solitary Wave Solution of Fractional Kudryashov-Sinelshchikov Equation Describing Non-linear Wave Processes in a Liquid Containing Gas Bubbles, Applied Mathematics and Computation, 298 (2017), 1, pp. 1-12
- Ak, T., et al., Polynomial and Rational Wave Solutions of Kudryashov-Sinelshchikov Equation and Numerical Simulations for Its Dynamic Motions, Journal of Applied Analysis and Computation, 5 (2020), 10, pp. 2145-2162
- Guner, O., et al., Dark Soliton and Periodic Wave Solutions of Non-linear Evolution Equations, Advances in Difference Equations, 1 (2013), 2013, pp. 1-11
- Qian, M. Y., He, J. H., Collection of Polymer Bubble as a Nanoscale Membrane, Surfaces and Interfaces, 28 (2022), Dec., 101665
- He, J. H., et al., The Maximal Wrinkle Angle During the Bubble Collapse and Its Application to the Bubble Electrospinning, Frontiers in Materials, 8 (2022), Feb., 800567
- Lin, L., et al., Fabrication of PVDF/PES Nanofibers with Unsmooth Fractal Surfaces by Electrospinning: A General Strategy and Formation Mechanism, Thermal Science, 25 (2021), 2, pp. 1287-1294
- Anjum, N., Ain, Q.T., Application of He's Fractional Derivative and Fractional Complex Transform for Time Fractional Camassa-Holm Equation, Thermal Science, 24 (2020), 5, pp. 3023-3030
- Wang, K. L., Yao, S. W., He's Fractional Derivative for the Evolution Equation, Thermal Science, 24 (2020), 4, pp. 2507-2513
- Lu, D. C., et al., Numerical Solutions of Coupled Non-linear Fractional KdV Equations Using He's Fractional Calculus, International Journal of Modern Physics B, 35 (2021), 3, 2150023
- Tian, D., et al. Fractal N/MEMS: From Pull-in Instability to Pull-in Stability, Fractals, 29 (2021), 2, 2150030
- He, C. H., et al., A Fractal Model for the Internal Temperature Response of a Porous Concrete, Applied and Computational Mathematics, 21 (2022), 1, pp. 71-77
- He, C. H., A Variational Principle for a Fractal Nano/Microelectromechanical (N/MEMS) System, International Journal of Numerical Methods for Heat & Fluid Flow, 33 (2022) 1, pp. 351-359
- He, C. H., et al., A Modified Frequency-Amplitude Formulation for Fractal Vibration Systems, Fractals, 30 (2022), 3, 2250046
- Anjum, N., et al., Two-Scale Fractal Theory for the Population Dynamics, Fractals, 29 (2021), 7, 2150182
- He, J. H., et al., Evans Model for Dynamic Economics Revised, AIMS Mathematics, 6( 2021), 9, pp. 9194-9206
- He, J. H., El-Dib, Y. O., Homotopy Perturbation Method with Three Expansions for Helmholtz-Fangzhu Oscillator, International Journal of Modern Physics B, 35 (2021), 24, 2150244
- He, C. H., El-Dib, Y.O.,A Heuristic Review on the Homotopy Perturbation Method for Non-Conservative Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 41(2022), 2, pp. 572-603
- Mahmoudi, Y., A New Modified Adomian Decomposition Method for Solving a Class of Hypersingular Integral Equations of Second Kind, Journal of Computational and Applied Mathematics, 255 (2014), 1, pp. 737-742
- Wazwaz, A. M., Adomian Decomposition Method for a Reliable Treatment of the Emden-Fowler Equation, Applied Mathematics and Computation, 2 (2005), 161, pp. 543-560
- Elsaid, A., Fractional Differential Transform Method Combined with the Adomian Polynomials, Applied Mathematics and Computation, 218 (2012), 12, pp. 6899-6911
- Wang, P., et al., Study on Vibration Response of a Non-Unform Beam with Non-linear Boundary Condition, Facta Universitatis Series: Mechanical Engineering, 19 (2021), 4, pp. 781-804
- Nadeem, M., He, J. H., He-Laplace Variational Iteration Method for Solving the Non-linear Equations Arising in Chemical Kinetics and Population Dynamics, Journal of Mathematical Chemistry, 59 (2021), 5, pp. 1234-1245
- Mishra, H. K., Tripathi, R., Homotopy Perturbation Method of Delay Differential Equation Using He's Polynomial with Laplace Transform, Proceedings of the National Academy of Sciences India Section A, 90 (2020), 2, pp. 289-298
- Ain, Q. T., He, J. H., On Two-scale Dimension and its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
- He, J. H., Seeing with a Single Scale is Always Unbelieving: From Magic to Two-Scale Fractal, Thermal Science, 25 (2021), 2, pp. 1217-1219
- Ain, Q. T., et al., The Fractional Complex Transform: A Novel Approach to the Time-Fractional Schrodinger Equation, Fractals, 28 (2020), 7, 2050141
- He, J. H., El-Dib, Y. O., A Tutorial Introduction to the Two-scale Fractal Calculus and its Application to the Fractal Zhiber-Shabat Oscillator, Fractals, 29 (2021), 8, 2150268
- He, J. H., Qian, M. Y., A Fractal Approach to the Diffusion Process of Red Ink in a Saline Water, Thermal Science, 26 (2022), 3B, pp. 2447-2451
- Qian, M. Y., He, J. H., Two-Scale Thermal Science for Modern Life - Making the Impossible Possible, Thermal Science, 26 (2022), 3B, pp. 2409-2412
- Yau, H. T., Generalized Projective Chaos Synchronization of Gyroscope Systems Subjected to Dead-Zone Non-linear Inputs, Physics Letters A, 372 (2008), 14, pp. 2380-2385
- Lin, J. S., et al., Synchronization of Unidirectional Coupled Chaotic Systems with Unknown Channel Time-Delay: Adaptive Robust Observer-Based Approach, Chaos, Solitons and Fractals, 26 (2005), 3, pp. 971-978
- Chen, C. L., et al., Terminal Sliding Mode Control for Aeroelastic Systems, Nonlinear Dynamics, 70 (2012), 3, pp. 2015-2026
- Yau, H. T., Yan, J. J., Adaptive Sliding Mode Control of a High-Precision Ball-Screw-Driven Stage, Nonlinear Analysis B, 10 (2009), 3, pp. 1480-1489