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<p>Higher-order nonlinear partial differential equations (PDEs) provide a powerful framework for modeling a wide range of complex phenomena. In this study, we investigate a general class of fourth-order nonlinear reaction-diffusion equations, focusing on the inverse problem of determining a time-dependent potential control term from an additional boundary measurement of the amplitude at the left boundary. The existence and uniqueness of the solution are established via the contraction mapping principle, providing a theoretical foundation for the control of nonlinear reaction-diffusion processes. For numerical implementation, the inverse problem is reformulated as a nonlinear least-squares minimization problem with bounded constraints on the unknown coefficient, and Tikhonov regularization is employed to ensure stability. The forward problem is discretized using the Crank–Nicolson finite-difference scheme, while the inverse problem is solved iteratively using MATLAB’s built-in routine <i>lsqnonlin</i>. Numerical outcomes for benchmark test examples are presented to demonstrate the accuracy and effectiveness of the proposed approach.</p>

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Identification of the purely time-dependent potential control term in a fourth-order nonlinear reaction-diffusion equation

  • 期刊论文

  • 发表日期：

    2026年5月2日

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  • M. Alosaimi,
  • I. Tekin

摘要

Higher-order nonlinear partial differential equations (PDEs) provide a powerful framework for modeling a wide range of complex phenomena. In this study, we investigate a general class of fourth-order nonlinear reaction-diffusion equations, focusing on the inverse problem of determining a time-dependent potential control term from an additional boundary measurement of the amplitude at the left boundary. The existence and uniqueness of the solution are established via the contraction mapping principle, providing a theoretical foundation for the control of nonlinear reaction-diffusion processes. For numerical implementation, the inverse problem is reformulated as a nonlinear least-squares minimization problem with bounded constraints on the unknown coefficient, and Tikhonov regularization is employed to ensure stability. The forward problem is discretized using the Crank–Nicolson finite-difference scheme, while the inverse problem is solved iteratively using MATLAB’s built-in routine lsqnonlin. Numerical outcomes for benchmark test examples are presented to demonstrate the accuracy and effectiveness of the proposed approach.