article · Journal of low frequency noise, vibration and active control
This research evaluates computational methods for solving fourth-order time-fractional Cahn-Hilliard equations, which model complex phase separation phenomena. The evaluation tests the Tantawy Technique alongside the variational iteration transform method and the homotopy perturbation transform method, both integrated with the Yang transform framework. These iterative approaches produce convergent series solutions for complex nonlinear problems. Testing on two nonlinear scenarios confirms the accuracy and numerical stability of the methods through direct comparisons with exact integer-order solutions and absolute error calculations. The findings show that altering fractional parameters directly influences the approximation profiles, with fractional solutions smoothly aligning with exact integer solutions as the order transitions. Overall, the studied analytical techniques provide computationally efficient, user-friendly tools that simplify the application of fractional calculus to complex physical models across science and engineering disciplines.
Nonlinear fractional differential equations describe complex physical behaviours such as phase separation in materials science and fluid mechanics. Solving these equations is often computationally demanding. Demonstrating fast, stable, and accurate iterative mathematical techniques simplifies the use of fractional calculus, allowing researchers and engineers to model intricate dynamic processes with greater precision and less computational expense.
The research represents early-stage theoretical and numerical development. By offering computationally efficient solvers, these techniques could eventually be integrated into scientific computing software, engineering simulation platforms, or materials design suites. Potential users include software developers and computational engineers modelling complex materials. However, the abstract demonstrates method validation on mathematical test cases, indicating that direct commercial deployment requires further integration into applied software tools.
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This investigation employs the “Tantawy Technique,” a novel, highly accurate, and fast technique, to solve and analyze the fourth-order time-fractional Cahn–Hilliard (TFCH) models. We also implement the variational iteration transform method (VITM) and the homotopy perturbation transform method (HPTM) within the Yang transform framework to analyze the fourth-order TFCH models. These iterative procedures enable the acquisition of solutions in a convergent series. We focus on two nonlinear issues in the current investigation to demonstrate the validity and accuracy of the suggested approaches. We check the derived approximations against the exact solutions for the integer cases and then find their absolute error to ensure their accuracy and stability. The present approaches show how different fractional orders affect the profile of the derived approximations. The presented issues demonstrate the precision and effectiveness of the suggested techniques in tackling strong nonlinear fractional differential equations. Implementing the suggested methodologies indicates that as the value of the fractional parameter transitions from fractional to integer order, the result approaches the exact solution for the integer case. These methods also make it straightforward to use fractal calculus in real life. The numerical analysis results demonstrate that the proposed iterative techniques are viable and user-friendly tools with high computational efficiency when used in various physical models in science and engineering.
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DOI: 10.1177/14613484251322240
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