Hyperbolic partial differential equations from the point of view of shearlet transforms

Subject Areas : Partial differential equations

R. Raisi Tousi ¹ , M. Aminkhah ²

1 -
2 -

Received: 2024-10-01 Accepted : 2024-12-30 Published : 2025-01-01

Keywords: Continuous shearlet system, hyperbolic partial differential equation, complex coefficients,

Abstract :

It is shown that a nonlinear partial differential equation with complex coefficients can be obtained through continuous shearlet transforms. We introduce a shearlet for constructing a nonlinear partial differential equation with complex coefficients and show that the initial value obtained from the continuous shearlet transform coincides with solutions of this equation.

References:

[1] S. Alinhac, Hyperbolic partial differential equations, Springer Science & Business Media, 2009.

[2] M. Amin Khah, A. Askari Hemmat, R. Raisi Tousi, Integral representation for solutions of the wave equation by shearlets, Optik. 127 (2016), 10554-10560.

[3] M. Amin Khah, R. Raisi Tousi, Numerical solutions of the wave equation using time-independent shearlet coefficients, Optik. (2021), 241:165914.

[4] C. Cattani, Harmonic wavelets towards the solution of nonlinear PDE, Comput. Math. Appl. 50 (8) (2005), 1191-1210.

[5] S. Dahlke, G. Steidl, G. Teschke, The continuous shearlet transform in arbitrary space dimensions, J. Fourier Anal. Appl. 16 (3) (2010), 340-364.

[6] J. Flaherty, Problems and techniques, SIAM Review. 47 (1) (2005), 41-42.

[7] P. Grohs, A. Obermeier, Optimal adaptive ridgelet schemes for linear advection equations, Appl. Comput. Harm. Anal. 41 (3) (2016), 768-814.

[8] M. Haase, J. Widjajakusuma, R. Bader, Scaling Laws and Frequency Decomposition from Wavelet Transform Maxima Lines and Ridges, Emergent Nature (2001), 365-374.

[9] H. Ishii, P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Diff. Equ. 83 (1) (1990), 26-78.

[10] B. J. Lewis, E. N. Onder, A. Prudil, Advanced Mathematics for Engineering Students: The Essential Toolbox, Butterworth-Heinemann, 2021.

[11] E. B. Postnikov, Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations, Comp. Math. Math. Phys. 46(1) (2006), 73-78.

[12] H. O. Sidi, M. A. Zaky, W. Qiu, A. S. Hendy, Identification of an unknown spatial source function in a multidimensional hyperbolic partial differential equation with interior degeneracy, Appl. Nume. Math. 192 (2023), 1-18.

[13] C. Tian, K. C. Chang, J. Chen, Application of hyperbolic partial differential equations in global optimal scheduling of UAV, Alex. Eng. J. 59 (4) (2020), 2283-2289.