A Simple Approach for the Fractal Riccati Differential Equation

Document Type : Research Paper

Author

School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo, 454003, China‎

Abstract

In this paper, a fractal modification of the Riccati differential equation is presented, and the two-scale transform method combined with Taylor series is used to solve the equation. Two examples are given to verify the correctness and effectiveness of the proposed method.

Keywords

Main Subjects

[1] Reid, W.T., Riccati Differential Equations, Academic Press, New York, 1972.
[2] Dehghan, M., Taleei, A., A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Computer Physics Communications, 181, 2010, 43–51.
[3] Mukherjee, S., Roy, B., Solution of Riccati equation with variable co-efficient by differential transform method, International Journal of Nonlinear Science, 14(2), 2012, 251–256.
[4] Wang, K.J. et al., The transient analysis for zero-input response of fractal RC circuit based on local fractional derivative, Alexandria Engineering Journal, 2020, https://doi.org/10.1016/j.aej.2020.08.024.
[5] Goswami, A., Singh, J., Kumar, D., et al., An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A, 524, 2019, 563-575.
[6] Kumar, D., Singh, J., Baleanu, D., On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Mathematical Methods in the Applied Sciences, 43(1), 2019, 443-457.
[7] Wang, K.J., On a High-pass filter described by local fractional derivative, Fractals, 2020, 28(3), 2050031.
[8] Wang, K.J., Sun, H.C., Cui, Q.C., The fractional Sallen-Key filter described by local fractional derivative, IEEE Access, 8, 2020, 166377-166383.
[9] Wang, K.J., et al., A a-order R-L high-pass filter modeled by local fractional derivative, Alexandria Engineering Journal, 59(5), 2020, 3244-3248.
[10] He, J.H., A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals, 28(2), 2020, 2050024.
[11] 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(2A), 2020, 659-681.
[12] He, J.H., Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves, Journal of Applied and Computational Mechanics, 6(4), 2020, 735-740.
[13] Wang, K.L., Wang, K.J., He, C.H., Physical insight of local fractional calculus and its application to fractional KdV-Burgers-Kuramoto equation, Fractals, 27(7), 2019, 1950122.
[14] Wang, K.L., Wang, K.J., A Modification of the Reduced Differential Transform Method for Fractional Calculus, Thermal Science, 22(4), 2018, 1871-1875.
[15] He, J.H., A tutorial review on fractal spacetime and fractional calculus, International Journal of Theoretical Physics, 53(11), 2014, 3698-3718.
[16] He, J.H., Fractal calculus and its geometrical explanation, Results in Physics, 10, 2018, 272–276.
[17] He, J.H., Li, Z.-B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16(2), 2012, 331-334.
[18] Ain, Q.T., He, J.H., On two-scale dimension and its applications, Thermal Science, 23(3B), 2019, 1707-1712.
[19] He, J.H., Ji, F.Y., Two-scale mathematics and fractional calculus for thermodynamics, Thermal Science, 23(4), 2019, 2131-2133.
[20] He, C.H., Shen, Y., Ji, F.Y., He, J.H., Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 28(1), 2020, 2050011.
[21] He, J.H., Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Engineering Journal, 2020, https://doi.org/10.1016/j.asej.2020.01.016.
[22] He, J.H., A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes, Journal of Electroanalytical Chemistry, 2019, 854, 113565.