Thermal Science

International Scientific Journal

Find this paper on

Qiaoling Chen

A VARIATIONAL PRINCIPLE FOR FRACTAL KLEIN-GORDON EQUATION

ABSTRACT

This paper studies the Klein-Gordon equation and two modifications in an infinite Cantor set and a fractal space-time. Their variational formulations are established and discussed, and the spatio-temporal discontinuity requires both spatio-fractal derivative and temporal fractal derivative for practical applications. Some basic properties of the local fractional derivative and the two-scale fractal derivative are elucidated, and the derivation of the Euler-Lagrange equation is illustrated.

KEYWORDS

variational theory, fractional calculation, two-scale fractal theory

PAPER SUBMITTED: 2021-12-20

PAPER REVISED: 2022-07-20

PAPER ACCEPTED: 2022-07-21

PUBLISHED ONLINE: 2023-06-11

DOI REFERENCE: https://doi.org/10.2298/TSCI2303803C

CITATION EXPORT: view in browser or download as text file

THERMAL SCIENCE YEAR 2023, VOLUME 27, ISSUE No. 3, PAGES [1803 - 1810]

REFERENCES

[1] Aryan, S., Existence of Two-Solitary Waves with Logarithmic Distance for the Non-linear Klein-Gordon Equation, Communications in Contemporary Mathematics, 24 (2022), 1, 2050091, 10.1142/s0219199720500911

[2] Sun, J. S., Approximate Analytical Solution of the Fractal Klein-Gordon Equation, Thermal Science, 25 (2021), 2, pp. 1489-1494, 10.2298/tsci200301051s

[3] He, J. H., El-Dib, Y. O., The Reducing Rank Method to Solve Third-Order Duffing Equation with the Homotopy Perturbation, Numerical Methods for Partial Differential Equations, 37 (2021), 2, pp. 1800- 1808, 10.1002/num.22609

[4] He, J. H., El-Dib, Y. O., The Enhanced Homotopy Perturbation Method for Axial Vibration of Strings, Facta Universitatis Series: Mechanical Engineering, 19 (2021), 4, pp. 735 - 750, 10.22190/fume210125033h

[5] Feng, G. Q., He's Frequency Formula to Fractal Undamped Duffing Equation, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 4, pp. 1671-1676, 10.1177/1461348421992608

[6] Wang, K. L., Wei, C. F., A Powerful and Simple Frequency Formula to Non-linear Fractal Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 3, pp. 1373-1379, 10.1177/1461348420947832

[7] He, J. H., et al., Forced Non-Linear Oscillator in a Fractal Space, Facta Universitatis Series: Mechanical Engineering, 20 (2022), 1, pp. 1-20, 10.22190/fume220118004h

[8] He, C. H., Liu, C., A Modified Frequency-Amplitude Formulation for Fractal Vibration Systems, Fractals, 30 (2022), 3, 2250046, 10.1142/s0218348x22500463

[9] He, J.-H., et al. Periodic Property and Instability of a Rotating Pendulum System. Axioms, 10 (2021), 3, 191, 10.3390/axioms10030191

[10] He, C. H., et al., Hybrid Rayleigh-Van der Pol-Duffing Oscillator (HRVD): Stability Analysis and Controller, Journal of Low Frequency Noise, Vibration & Active Control, 41 (2022), 1, pp. 244-268, 10.1177/14613484211026407

[11] Wang, K. J., Wang, G. D., Gamma Function Method for the Non-Linear Cubic-Quintic Duffing Oscillators, Journal of Low Frequency Noise, Vibration & Active Control, 41 (2022), 1, pp. 216-222, 10.1177/14613484211044613

[12] Ma, H. J., Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems, Facta Universitatis Series: Mechanical Engineering, 20 (2022), 2, pp. 445-455, 10.22190/fume220420023m

[13] Wang, K. L., Wei, C. F., A Powerful and Simple Frequency Formula to Non-Linear Fractal Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 3, pp. 1373-1379

[14] Cuzinatto, R. R., et al. Non-Commutativity and Non-Inertial Effects on a Scalar Field in a Cosmic String Space-Time: I. Klein-Gordon Oscillator, Classical and Quantum Gravity, 39 (2022), 7, 075006, 10.1088/1361-6382/ac51bb

[15] Shen, Y., et al., Convergence of Adaptive Non-Conforming Finite Element Method for Stokes Optimal Control Problems, Journal of Computational and Applied Mathematics, 412 (2022), Oct., 114336, 10.1016/j.cam.2022.114336

[16] 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, 10.1108/hff-03-2022-0191

[17] He, J.-H., A Fractal Variational Theory for One-Dimensional Compressible Flow in a Microgravity Space, Fractals, 28 (2020), 2, 20500243, 10.1142/s0218348x20500243

[18] He, J.-H., On the Fractal Variational Principle for the Telegraph Equation, Fractals, 29 (2021), 1, 2150022, 10.1142/s0218348x21500225

[19] Wang, Y., et al., A Variational Formulation Anisotropic Wave Travelling in a Porous Medium, Fractals, 27 (2019), 4, 1950047, 10.1142/s0218348x19500476

[20] Wang, Y., et al., A Fractal Derivative Model for Snow's Thermal Insulation Property, Thermal Science. 23 (2019), 4, pp. 2351-2354, 10.2298/tsci1904351w

[21] Wang, K. J., Generalized Variational Principle and Periodic Wave Solution to the Modified Equal Width-Burgers Equation in Non-linear Dispersion Media, Physics Letters A, 419 (2021), 17, 127723, 10.1016/j.physleta.2021.127723

[22] Wang, K. J., Zhang, P. L., Investigation of the Periodic Solution of the Time-Space Fractional Sasa-Satsuma Equation Arising in the Monomode Optical Fibers, EPL, 137 (2022), 6, 62001, 10.1209/0295-5075/ac2a62

[23] Wang, K. J., Zhu, H. W., Periodic Wave Solution of the Kundu-Mukherjee-Naskar Equation in Birefringent Fibers via the Hamiltonian-Based Algorithm, EPL, 139 (2021), 3, 35002, 10.1209/0295-5075/ac3d6b

[24] Wang, K J., Wang, J. F., Generalized Variational Principles of the Benney-Lin Equation Arising in Fluid Dynamics, EPL, 139 (2021), 3, 39006, 10.1209/0295-5075/ac3cce

[25] Wang, K. J., Liu, J. H., Periodic Solution of the Time-Space Fractional Sasa-Satsuma Equation in the Monomode Optical Fibers by the Energy Balance Theory, EPL, 138 (2022), 2, 25002, 10.1209/0295-5075/ac5c78

[26] Wang, K. L., Exact Solitary Wave Solution for Fractal Shallow Water Wave Model by He's Variational Method, Modern Physics Letters B, 36 (2022), 7, 2150602, 10.1142/s0217984921506028

[27] Wang, K. L., Solitary Wave Solution of Non-linear Bogoyavlenskii System by Variational Analysis Method, International Journal of Modern Physics B, 36 (2022), 2, 2250015, 10.1142/s0217979222500151

[28] Wang, K. L., New Variational Theory for Coupled Non-linear Fractal Schrodinger System, International Journal of Numerical Methods for Heat & Fluid Flow, 32 (2022), 2, pp. 589-597, 10.1108/hff-02-2021-0136

[29] Khan, Y., A Variational Approach for Novel Solitary Solutions of FitzHugh-Nagumo Equation Arising in the Non-linear Reaction-Diffusion Equation, International Journal of Numerical Methods for Heat and Fluid Flow, 31 (2020), 4, 1104-1109, 10.1108/hff-05-2020-0299

[30] Khan, Y., Fractal Modification of Complex Ginzburg-Landau Model Arising in the Oscillating Phenomena, Results in Physics, 18 (2020), Sept., 103324, 10.1016/j.rinp.2020.103324

[31] Zuo, Y.-T., Liu, H.-J., Fractal Approach to Mechanical and Electrical Properties of Graphene/Sic Composites, Facta Universitatis-Series Mechanical Engineering, 19 (2021), 2, pp. 271-284, 10.22190/fume201212003z

[32] Tian, D., He, C. H., A Fractal Micro-Electromechanical System and Its Pull-In Stability, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 3, pp. 1380-1386, 10.1177/1461348420984041

[33] Cao, X.-Q., et al. Variational Theory for 2+1 Dimensional Fractional Dispersive Long Wave Equations, Thermal Science, 25 (2021), 2B, pp. 1277-1285, 10.2298/tsci200301023c

[34] Alex, E. Z., et al., Equivalent Power-Form Representation of the Fractal Toda Oscillator, Fractals, 29 (2020), 2, 21500341

[35] Alex, E. Z., et al., He's Frequency-Amplitude Formulation for Non-Linear Oscillators Using Jacobi Elliptic Functions, Journal of Low Frequency Noise Vibration and Active Control, 29 (2021), 2, 2150034

[36] He, J.-H., Seeing with a Single Scale is Always Unbelieving: From magic to two-scale fractal, Thermal Science, 25 (2021), 2B, pp. 1217-1219, 10.2298/tsci2102217h

[37] He, J.-H. When Mathematics Meets Thermal Science: The Simpler is the Better, Thermal Science, 25 (2021), 3, pp. 2039-2042, 10.2298/tsci200715132h

[38] Jia, Z. J., et al., Variational Principle for Unsteady Heat Conduction Equation, Thermal Science, 18 (2014), 3 , pp. 1045-1047, 10.2298/tsci140108027j

[39] Wang, K. L., Wang, K. J., A New Analysis for Klein-Gordon Model with Local Fractional Derivative, Alexandria Engineering Journal, 59 (2020), 5, pp. 3313-3309, 10.1016/j.aej.2020.04.040

[40] Wang, K. J., On New Abundant Exact Traveling Wave Solutions to the Local Fractional Gardner Equation Defined on Cantor Sets, Mathematical Methods in the Applied Sciences, 45 (2022), 4, pp. 1904-1915, 10.1002/mma.7897

[41] Yang, X. J., et al., On Local Fractional Operators View of Computational Complexity: Diffusion and Relaxation Defined on Cantor Sets, Thermal Science, 20 (2016), Suppl. 3, pp. S755-S767, 10.2298/tsci16s3755y

[42] Yang, X. J., et al., Local Fractional Similarity Solution for the Diffusion Equation Defined on Cantor Sets, Applied Mathematics Letters, 47 (2015), Sept., pp. 54-60, 10.1016/j.aml.2015.02.024

[43] 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, 10.1142/s0217979221502143

[44] Anjum, N., et al. Two-Scale Fractal Theory for the Population Dynamics, Fractals, 29 (2021), 7, 2150182, 10.1142/s0218348x21501826

[45] Wei, C. F., Two-Scale Transform for 2-D Fractal Heat Equation in a Fractal Space, Thermal Science, 25 (2021), 3, pp. 2339-2345, 10.2298/tsci190918124w

[46] 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, 10.2298/tsci2203409q

[47] He, J. H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199, 10.1142/s0218348x21501991

[48] He, J. H., et al., On a Strong Minimum Condition of a Fractal Variational Principle, Applied Mathematics Letters, 119 (2021), Sept., 107199, 10.1016/j.aml.2021.107199

[49] Wang, S. Q., He, J. H., Variational Iteration Method for Solving Integro-Differential Equations, Physics letters A, 367 (2007), 3, pp. 188-191, 10.1016/j.physleta.2007.02.049

[50] Wang, S. Q., A Variational Approach to Non-Linear Two-Point Boundary Value Problems, Computers & Mathematics with Applications, 58 (2009), 11, pp. 2452-2455, 10.1016/j.camwa.2009.03.050

[51] Yu, W., et al., Tensorizing GAN with High-Order Pooling for Alzheimer's Disease Assessment, IEEE Transactions on Neural Networks and Learning Systems, 33 (2021), 9, 4945-4959, 10.1109/tnnls.2021.3063516

[52] You, S., et al., Fine Perceptive Gans for Brain MR Image Super-Resolution in Wavelet Domain, IEEE transactions on neural networks and learning systems, On-line first, 2022.3153088, 2022, 10.1109/tnnls.2022.3153088

[53] Hu, S., et al., Bidirectional Mapping Generative Adversarial Networks for Brain MR to PET Synthesis, IEEE Transactions on Medical Imaging, 41 (2021), 1, pp. 145-157, 10.1109/tmi.2021.3107013

[54] Yu, W., et al., Morphological Feature Visualization of Alzheimer's Disease via Multidirectional Perception GAN, IEEE Transactions on Neural Networks and Learning Systems, On-line first, , 2021, 10.1109/TNNLS.2021.3118369

PDF VERSION [DOWNLOAD]

A variational principle for fractal Klein-Gordon equation

© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence