The Klein-Gordon problem (KGP) is gaining attention within scientific circles for its ability to model various complex physical phenomena. Researchers at the forefront of this field have now proposed an innovative algorithm known as the Confluent Bernoulli approach with Residual Power Series Scheme (CBCA-RPSS), which offers promising applications for solving the fractional non-linear KGP.
This algorithm stands out by leveraging the principles of fractional calculus—a powerful branch of mathematics devoted to analyzing integrals and derivatives of non-integer orders. Over the years, fractional calculus has emerged as an effective tool across fields ranging from mathematical physics to engineering, providing insights for phenomena best described by fractional partial differential equations (FPDEs).
Driven by the increasing recognition of fractional calculus' potential, the researchers conducted rigorous analyses of their proposed method, evaluating its convergence, uniqueness, and error characteristics. They found substantial improvements over several well-established numerical techniques through their experiments.
Specifically, the numerical simulations indicated the method's efficiency and superiority. By generating codes implemented within the MATLAB R2017b platform, the researchers charted error norms, demonstrating the reliability of CBCA-RPSS against other existing techniques. Their insights indicate this approach can handle both linear and nonlinear fractional models effectively.
Among the compelling reasons for focusing on the KGP is its diverse application spectrum—as it can model everything from nonlinear wave propagation to plasma physics and quantum field theories. Given this background, the researchers emphasized the need for advanced solutions, hence their development of CBCA-RPSS.
They highlighted the algorithm's foundation, utilizing concepts from fractional calculus and Bernoulli polynomials—both significant players within the mathematical machinery employed for the analysis of KGP. The inventions drew comparisons to traditional methodologies, such as Adomian decomposition and finite element methods, showcasing how CBCA-RPSS circumvents limitations associated with earlier techniques.
The introduction of CBCA-RPSS provides the scientific community with an accurate mechanism capable of addressing the intricacies of KGP. The research team detailed their unique approach, which transforms the fractional KGP using confluent Bernoulli polynomials and, through systematic substitution, converts the problem to yield algebraic equations solvable via RPSS.
With several numerical examples presented, the researchers illustrated error norms across various scenarios, underpinning the practical effectiveness of their approach. Detailed comparisons with other methodologies confirmed significant advantages, including reduced computational costs and improved accuracy rates.
Through visual aids, including error graphs and comparative data tables, the findings demonstrate the practicality of CBCA-RPSS, inviting future applications stretching beyond KGP, such as those connected to complex solitonic behaviors.
Concluding their work, the researchers advocate for the broader adoption of CBCA-RPSS across nonlinear partial differential equations, emphasizing its role as a straightforward and effective solution technique. This breakthrough opens pathways toward resolving complex equations previously deemed challenging or computationally prohibitive.
