## THERMAL SCIENCE

International Scientific Journal

### VARIATIONAL ITERATION METHOD FOR TWO FRACTIONAL SYSTEMS WITH BOUNDARY CONDITIONS

**ABSTRACT**

Under investigation in this paper are two local fractional partial differential systems, one is the homogeneous linear partial differential system with initial values, and the other is the inhomogeneous non-linear partial differential system with initial and boundary values. To solve these two local fractional systems, we employ the local fractional variational iteration method and obtain exact solutions. It is shown that the method provides an effective mathematical tool for solving linear and non-linear local fractional partial differential systems with initial and boundary values.

**KEYWORDS**

PAPER SUBMITTED: 2020-10-04

PAPER REVISED: 2021-10-01

PAPER ACCEPTED: 2021-10-01

PUBLISHED ONLINE: 2022-07-16

**THERMAL SCIENCE** YEAR

**2022**, VOLUME

**26**, ISSUE

**Issue 3**, PAGES [2653 - 2661]

- Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, Cal., USA, 1999
- He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- He, J. H., Fractal Calculus and its Geometrical Explanation, Results in Physics, 10, (2018), 1, pp. 272-276
- He, J. H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199
- He, J. H., Maximal Thermo-geometric Parameter in a Non-linear Heat Conduction Equation, Bulletin of the Malaysian Mathematical Sciences Society, 39 (2016), 2, pp. 605-608
- 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 (2021), 1, pp. 244-268
- Tian, D., et al., Fractal N/MEMS: From Pull-in Instability to Pull-in Stability, Fractals, 29 (2021), 2, 2150030
- 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
- Kolwankar, K. M., Gangal, A. D., Fractional Differentiability of Nowhere Differentiable Functions and Dimensions, Chaos, 6 (1996), 4, pp. 505-513
- Yang, X. J., et al., Local Fractional Integral Transforms and their Applications, Elsevier, London, UK, 2015
- Zhang, S., et al., Fractional Derivative of Inverse Matrix and its Applications to Soliton Theory, Thermal Science, 24 (2020), 4, pp. 2597-2604
- Yang, Y. J., The Fractional Residual Method for Solving the Local Fractional Differential Equations, Thermal Science, 24, (2020), 4, pp. 2535-2542
- Yang, Y. J., A Local Fractional Variational Iteration Method for Laplace Equation within Local Frac-tional Operators, Abstract and Applied Analysis, 2013 (2014), Feb., ID 202650
- Zhang, S., Zhang, H. Q., Fractional Sub-Equation Method and its Applications to Non-linear Fractional PDEs, Physics Letters A, 375 (2011), 7, pp. 1069-1073
- Zhang, S., et al., Variable Separation Method for Non-linear Time Fractional Biological Population Model, International Journal of Numerical Methods for Heat and Fluid Flow, 25 (2015), 7, pp. 1531-1541
- Shi, D. D., Zhang, Y. F., Diversity of Exact Solutions to the Conformable Space-Time Fractional MEW Equation, Applied Mathematics Letters, 99 (2020), Jan., ID 105994
- Zhang, S., Hong, S. Y., Variable Separation Method for a Non-linear Time Fractional Partial Differential Equation with Forcing Term, Journal of Computational and Applied Mathematics, 339 (2018), Apr., pp. 297-305
- Xu, B., et al., Analytical Insights into Three Models: Exact Solutions and Non-linear Vibrations, Journal of Low Frequency Noise, Vibration & Active Control, 38 (2019), 3-4, pp. 901-913
- Zhang, S., et al., Bilinearization and Fractional Soliton Dynamics of Fractional Kadomtsev-Petviashvili Equation, Thermal Science, 23 (2019), 3, pp. 1425-1431
- Zhang, S., et al., Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Non-linear Systems: a Concrete Example, Complexity, (2019), Aug., ID 7952871
- He, J. H., Variational Iteration Method-a Kind of Non-linear Analytical Technique: Some Examples, In-ternational Journal of Non-Linear Mechanics, 34 (1999), 4, pp. 699-708
- He, J. H., Wu, X. H., Variational Iteration Method: New Development and Applications, Computers & Mathematics with Applications, 54 (2007), 7-8, pp. 894-881
- He, J. H., Wu, X. H., Variational Iteration Method: New Development and Applications, Computers & Mathematics with Applications, 54 (2007), 7-8, pp. 881-894
- Anjum, N. He, J. H., Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), Jun., pp. 134-138
- He, J. H., Variational Iteration Method - Some Recent Results and New Interpretations, Journal of Computational and Applied Mathematics, 207 (2007), 1, pp. 3-17
- He, J. H., et al., Approximate Periodic Solutions to Microelectromechanical System Oscillator Subject to Magnetostatic Excitation, Mathematical Methods in Applied Sciences, On-line first, doi.org/10. 1002/mma.7018, 2020
- Anjum, N., He, J. H., Analysis of Non-linear Vibration of Nano/Microelectromechanical System switch Induced by Electromagnetic Force Under Zero Initial Conditions, Alexandria Engineering Journal, 59 (2020), 6, pp. 4343-4352
- Yang, Y. J., The Local Fractional Variational Iteration Method a Promising Technology for Fractional Calculus, Thermal Science, 24 (2020), 4, pp. 2605-2614
- Wazwaz, A. M., The Variational Iteration Method for Solving Linear and Non-linear Systems of PDEs, Computers & Mathematics with Applications, 54 (2007), 7-8, pp. 895-902
- Wazwaz, A. M., The Variational Iteration Method: A Reliable Analytic Tool for Solving Linear and Non-linear Wave Equations, Computers & Mathematics with Applications, 54 (2007), 7-8, pp. 926-932
- Tian, Y., Liu, J., A Modified Exp-Function Method for Fractional Partial Differential Equations, Ther-mal Science, 25 (2021), 2, pp. 1237-1241
- Tian, Y., Liu, J., Direct Algebraic Method for Solving Fractional Fokas Equation, Thermal Science, 25 (2021), 3, pp. 2235-2244
- Tian, Y., Wan, J. X., Exact Solutions of Space-Time Fractional 2+1 Dimensional Breaking Soliton equa-tion, Thermal Science, 25 (2021), 2, pp. 1229-1235
- Wang, K. J., On New Abundant Exact Traveling Wave Solutions to the Local Fractional Gardner Equa-tion Defined on Cantor Sets, Mathematical Methods in the Applied Sciences, 45 (2021), 4, pp. 1904-1919
- 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), Dec., 127723
- 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 (2021), 6, 62001
- Han, C., et al., Numerical Solutions of Space Fractional Variable-Coefficient KdV-Modified KdV Equa-tion by Fourier Spectral Method, Fractals, 29 (2021), 8, 21502467
- Dan, D. D., et al.,Using Piecewise Reproducing Kernel Method and Legendre Polynomial for Solving a Class of the Time Variable Fractional Order Advection-Reaction-Diffusion Equation, Thermal Science, 25 (2021), 2B, pp. 1261-1268
- Feng, G. Q., He's Frequency Formula to Fractal Undamped Duffing Equation, Journal of Low Frequen-cy Noise Vibration and Active Control, 40 (2021), 4, pp. 1671-1676