THERMAL SCIENCE
International Scientific Journal
ANALYTICAL STUDY OF FRACTAL MODIFIED DEGASPERIS-PROCESI EQUATION INVOLVING BETA-DERIVATIVE
ABSTRACT
This paper considers the fractal modified Degasperis-Procesi type equation involving a Beta-derivative as a generalized form of the standard ones. The approximate analytical solutions for the new model were obtained by employing the modified homotopy perturbation method coupled Laplace transformation, which is also called as He-Laplace method in literature. The presented example demonstrates the efficacy of the applied method in solving non-linear equations.
KEYWORDS
PAPER SUBMITTED: 2023-08-05
PAPER REVISED: 2024-06-15
PAPER ACCEPTED: 2024-06-15
PUBLISHED ONLINE: 2025-07-06
THERMAL SCIENCE YEAR
2025, VOLUME
29, ISSUE
Issue 3, PAGES [1739 - 1747]
- Wazwaz, A. M., Solitary Wave Solutions for Modified Forms of Degasperis-Procesi and Camassa-Holm Equations, Physics Letters A, 6 (2006), 352, pp. 500-504
- Lenells, J., The Degasperis-Procesi Equation on the Half-Line, Non-linear Analysis: Theory, Methods and Applications, 76 (2013), 12, pp. 122-139
- Ma, H. C., et al., New Exact Traveling Wave Solutions for the Modified form of Degasperis-Procesi Equation, Applied Mathematics and Computation, 2 (2008), 203, pp. 792-798
- Wei, M., Bifurcations and Exact Traveling Wave Solutions for a Modified Degasperis-Procesi Equation, Advances in Difference Equations, 2019 (2019), 126
- Chen, C., Tang, M., A New Type of Bounded Waves for Degasperis-Procesi Equation, Chaos, Solitons and Fractals, 3 (2006), 27, pp. 698-704
- Zong, X., Zhao, Y., Stabilization of a New Periodic Integrable Equation: The Degasperis-Procesi Equation, Non-linear Analysis: Theory, Methods and Applications, 11 (2007), 67, pp. 3167-3175
- Kolev, B., Some geometric Investigations on the Degasperis-Procesi Shallow Water Equation, Wave Motion, 6 (2009), 46, pp. 412-419
- Vakhnenko, V. O., Parkes, E. J., Periodic and Solitary-Wave Solutions of the Degasperis-Procesi Equation, Chaos, Solitons and Fractals, 5 (2004), 20, pp. 1059-1073
- Lundmark, H., Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation, Journal of Non-Linear Science, 17 (2007), 1, pp. 169-198
- Kilbas A. A., Marzan, S. A., Non-linear Differential Equations with the Caputo Fractional Derivative in the Space of continuously Differentiable Functions, Differential Equations, 1 (2005), 41, pp. 84-89
- He, J.-H., A Tutorial Review on Fractal Space-time and Fractional Calculus, International Journal of Theoretical Physics, 11 (2014), 53, pp. 3698-3718
- Chen, W., et al., Investigation on Fractional and Fractal Derivative Relaxation Oscillation Models, International Journal of Non-linear sciences and Numerical Simulation, 1 (2010), 11, pp. 3-10
- Fan, J., He, J.-H., Fractal Derivative Model for Air Permeability in Hierarchic Porous Media, Abstract and Applied Analysis, 2 (2015), 2012, pp. 97-112
- Li, X. M., et al., Variational Principle of the 2-D Steady Convection-Diffusion Equation with Fractal Derivatives, Thermal Science, 27 (2023), 3A, pp. 2049-2055
- He, J.-H., et al., Forced Non-Linear Oscillator in a Fractal Space. Facta Universitatis Series: Mechanical Engineering, 20 (2022), 1, pp. 1-20
- He, C. H., Liu, C., A Modified Frequency-Amplitude Formulation for Fractal Vibration Systems, Fractals, 30 (2022), 3, 2250046
- He, J.-H., et al., Pull-in Stability of a Fractal MEMS System and Its Pull-In Plateau, Fractals, 30 (2022), 9, 22501857
- He, J.-H., et al., Piezoelectric Biosensor Based on Ultrasensitive MEMS System, Sensors and Actuators: A. Physical, 376 (2024), 115664
- He, C. H., A Variational Principle for a fractal Nano/Microelectromechanical (N/MEMS) System, Int. J. Numer. Method. H., 33 (2023), 1, pp. 351-359
- He, J.-H., et al., Pull-Down Instability of the Quadratic Non-Linear Oscillators, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 2, pp. 191-200
- 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, 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, C. H., Liu, C., Fractal Approach to the Fluidity of a Cement Mortar, Non-linear Engineering-Modeling and Application, 11 (2022), 1, pp. 1-5
- He, J.-H., et al., A Fractal Modification of Chen-Lee-Liu Equation and Its Fractal Variational Principle, International Journal of Modern Physics B, 35 (2021), 21, 2150214
- 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
- Yepez-Martinez, H., et al., Beta-derivative and Sub-Equation Method Applied to the Optical Solitons in Medium with Parabolic Law Non-Linearity and higher Order Dispersion, Optik, 155 (2018), Feb., pp. 357-365
- Feng, G. Q., Niu, J. Y., He's Frequency Formulation for Non-Linear Vibration of a Porous Foundation with Fractal Derivative, GEM-International Journal on Geomathematics, 12 (2021), 1, 14
- 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
- He, C. H., et al., Hybrid Rayleigh-van der Pol-Duffing Oscillator: Stability Analysis and Controller, Journal of Low Frequency Noise Vibration and Active Control, 41 (2022), 1, pp. 244-268
- He, J.-H., et al., Good Initial Guess for Approximating Non-Linear Oscillators by the Homotopy Perturbation Method, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 1, pp. 21-29
- Anjum, N., et al., Free Vibration of a Tapered Beam by the Aboodh Transform-based Variational Iteration Method, Journal of Computational Applied Mechanics, 55 (2024), 3, pp. 440-450
- Zhang, B., et al., Homotopy Perturbation Method for modified Camassa-Holm and Degasperis-Procesi Equations, Physics Letters A, 11 (2008), 372, pp. 1867-1872
- Gupta, P. K., et al., Approximate Analytical Solution of the Time-Fractional Camassa-Holm, Modified Camassa-Holm, and Degasperis-Procesi Equations by Homotopy Perturbation Method, Scientia Iranica, 1 (2016), 23, pp. 155-165
- Salas, A. H., et al., New Solutions for the Modified Generalized Degasperis-Procesi Equation, Applied Mathematics and Computation, 215 (2009), 7, pp. 2608-2615
- Prakash, D. V., et al., An Efficient Computational Technique for Time-Fractional Modified Degasperis-Procesi Equation Arising in pRopagation of Non-Linear Dispersive Waves, Journal of Ocean Engineering and Science, 1 (2021), 6, pp. 30-39
- Singh, J., Gupta, A., Computational Analysis of Fractional Modified Degasperis-Procesi Equation with Caputo-Katugampola Derivative, AIMS Mathematics, 8 (2023), 1, pp. 194-212
- Celik, I., Jacobi Wavelet Collocation Method for the Modified Camassa-Holm and Degasperis-Procesi Equations, Engineering with Computers, 38 (2022), Suppl. 3, pp. S2271-S2287
- Chen, B., et al., He-Laplace Method for Time Fractional Burgers-type Equation, Thermal Science, 27 (2023), 3A, pp. 1947-1955
- Gurefe, Y., The Generalized Kudryashov Method for the Non-Linear Fractional Partial Differential Equations with the Beta-Derivative, Revista Mexicana de Fisica, 66 (2020), 6, pp. 771-781
- He, C. H., Liu, C., Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete's Strength, Facta Universitatis Series: Mechanical Engineering, 21 (2023), 1, pp. 137-150
- Nadeem, M., He, J.-H., The Homotopy Perturbation Method for Fractional Differential Equations: Part 2, Two-Scale Transform, International Journal of Numerical Methods for Heat and Fluid Flow, 32 (2022), 2, pp. 559-567
- He, J.-H., et al., Non-linear Instability of Two Streaming-Superposed Magnetic Reiner-Rivlin Fluidsby-He-Laplace Method, Journal of Electroanalytical Chemistry, 895 (2021), 115388
- Qayyum, M., et al., New Solutions of Fractional 4D Chaotic Financial Model with Optimal Control via He-Laplace Algorithm, Ain Shams Engineering Journal, 15 (2024), 3, 102503
- Anjum, N., et al., Li-He's Modified Homotopy Perturbation Method for Doubly-Clamped Electrically Actuated Microbeams-Based Microelectromechanical System, Facta Univ.-Ser. Mech., 19 (2021), 4, pp. 601-612
- Ghorbani, A., Beyond Adomian Polynomials: He Polynomials, Chaos, Solitons and Fractals, 39 (2009), 3, pp. 1486-1492
- Elias-Zuniga, A., et al., Analysis of Damped Fractal System Using the Ancient Chinese Algorithm and the Two-Scale Fractal Dimension Transform, Fractals, 30 (2022), 9, 22501730
- 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), 5A, pp. 3023-3030
- He, J.-H., et al., Geometrical Explanation of the Fractional Complex Transform and Derivative Chain Rule for Fractional Calculus, Physics Letters, 376 (2012), 4, pp. 257-259
- Wazwaz, A. M., New solitary Wave Solutions to the Modified Forms of Degasperis-Procesi and Camassa-Holm Equations, Applied Mathematics and Computation, 186 (2007), 1, pp. 130-141
- Song, Q. R., Zhang, J. G., He-transform: Breakthrough Advancement for the Variational Iteration Method, Frontiers in Physics, 12 (2024), 1411691
- He, J.-H., et al., Beyond Laplace and Fourier Transforms: Challenges and Future Prospects, Thermal Science, 27 (2023), 6B, pp. 5075-5089