Numerical Solution of the 2D Cauchy–Riemann System Using Classical and Quantum-Inspired Finite Difference and Crank–Nicolson Schemes

Authors

  • Mitat Uysal, Aynur Uysal

Keywords:

Cauchy–Riemann equations, Finite Difference, Crank–Nicolson, Quantum-Inspired Numerical Methods, Complex Analysis, Stability, PDE Discretization.

Abstract

The Cauchy–Riemann (CR) equations form the fundamental condition for analyticity in complex analysis and arise in potential theory, fluid mechanics, and electromagnetic field modeling. In this study, the two-dimensional Cauchy–Riemann system is solved numerically under prescribed Dirichlet boundary conditions using four approaches: (i) Finite Difference (FD), (ii) Quantum-Inspired Finite Difference (QI-FD), (iii) Crank–Nicolson (CN), and (iv) Quantum-Inspired Crank–Nicolson (QI-CN). Full mathematical derivations of discretization schemes are provided. The quantum-inspired schemes introduce amplitude-modulated update operators motivated by quantum probability dynamics. Comparative simulations demonstrate convergence behavior, stability properties, and error characteristics. Multiple graphical outputs including surface plots, contour maps, error heatmaps, and convergence curves are presented.

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References

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Published

26.02.2026

How to Cite

Mitat Uysal. (2026). Numerical Solution of the 2D Cauchy–Riemann System Using Classical and Quantum-Inspired Finite Difference and Crank–Nicolson Schemes. International Journal of Intelligent Systems and Applications in Engineering, 14(1s), 49–52. Retrieved from https://ijisae.org/index.php/IJISAE/article/view/8098

Issue

Vol. 14 No. 1s (2026)

Section

Research Article

License

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