Numerical estimation of the fractional Klein-Gordon equation with Discrete

We embark on a thorough analysis of a fractional model, concentrating our efforts on exploring the intricacies of the Klein -Gordon equation, within the framework of a specialized fractional operator. Our methodology is defined by the incorporation of the Modified Atangana Baleanu Caputo (MABC) derivative, representing an enhanced evolution of the original Atangana-Baleanu derivative. The core objective of our intricate investigation is to uncover the approximate solutions of the meticulously crafted model, employing the refined computational capabilities of Discrete Chebyshev Polynomials (DCPs) to facilitate our analytical endeavors. Laying the analytical groundwork, we meticulously derive a sequence of cutting -edge operational matrices, which are synthesized by amalgamating the intrinsic attributes of DCPs and the pertinent derivatives. This sophisticated development directs us towards an elaborate algebraic system, demanding precise and accurate resolution. The structured operational matrices play a crucial role in transforming the intricate fractional differential equations into more manageable algebraic equations, allowing the application of versatile numerical techniques to find solutions and making the entire process more approachable and conclusive. Our commitment to methodological rigor and computational precision is unwavering, ensuring the reliability and validity of the proposed methodology through extensive testing on diverse examples, revealing minimal errors in the outcomes. These results underscore the robust reliability and substantial effectiveness of the presented approach, thereby confirming its promising applicability in addressing similar fractional models of differential equations. The negligible discrepancies observed in the results serve as a testament to the potential widespread applicability of this methodology, offering substantial contributions to the existing scientific discourse and providing fertile ground for future research in the realm of fractional calculus and its associated fields. The minimal discrepancies detected in the outcomes exemplify the expansive applicability of this methodology, marking significant advancements in the scientific narrative and fostering opportunities for ensuing research in fractional calculus and related domains. This methodology can potentially be employed in areas such as quantum mechanics and signal processing, allowing for enhanced analysis and solutions of complex systems, thereby contributing to advancements in the development of more accurate models and simulations in these fields.