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Analytical transient analysis of temporal boundary value problems using the d'Alembert formula.

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    Temporal boundary value problems (TBVPs) are unbounded initial value problems with traveling wave solutions. Analytical transient wave solutions are derived using d'Alembert

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    Area of Science:

    • Electromagnetics and Wave Propagation
    • Mathematical Physics

    Background:

    • Temporal boundary value problems (TBVPs) are crucial for analyzing electromagnetic waves in time-varying media.
    • Existing methods often lack analytical solutions for transient wave behavior.

    Purpose of the Study:

    • To categorize TBVPs as unbounded initial value problems.
    • To develop an analytical method for evaluating transient wave expressions through temporal boundaries.

    Main Methods:

    • Classifying TBVPs within the framework of unbounded initial value problems.
    • Employing the d'Alembert formula on divided time subdomains.
    • Leveraging causality inherent to temporal systems.

    Main Results:

    • TBVPs are shown to possess traveling wave solutions.
    • Analytical transient wave expressions are derived for propagation across temporal boundaries.
    • The uniqueness of these analytical solutions for temporal systems is demonstrated.

    Conclusions:

    • The d'Alembert formula provides a powerful tool for solving TBVPs analytically.
    • Understanding the causality in TBVPs is key to their unique solution characteristics.
    • This work offers a foundational approach for analyzing transient electromagnetic waves in dynamic media.