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    Topological degree theory to investigate fractional order korteweg-de vries equation of porous media
    (2026-01-01)
    Ullah, Atta
    ;
    Khan, Zareen A.
    ;
    Fatima, Nahid
    ;
    Mlaiki, Nabil
    ;
    Jeelani, Mdi Begum
    The Korteweg-de Vries (KdV) equation is considered in this paper with fractional order derivative. We have considered the aforesaid equation under the non-singular derivative of fractional order introduced by Atangana-Baleanu-Caputo (ABC). We have implemented topological degree theory to deduce sufficient results for the existence and uniqueness of solution to the concerned problem. The concerned degree theory has been used to study a class of nonlinear integral equation. In this regard, we considered a general form for our proposed problem and then established appropriate conditions required for the existence and uniqueness of solution. Additionally, stability is an important requirement in investigating analytical or numerical solutions, therefore, we have attempted to derive some adequate results for the Hyers-Ulam-Type stability. Finally, for computing the approximate solution, we have used Laplace transform coupled with Adomian decomposition (LADM) to study the mentioned problem. Two concerted examples have been testified by the mentioned technique. Different surface plots have been presented to investigate the physical impact of fractional order, time and space variable on the propagation of wave solution.
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    Fractional-order Modeling and Optimal Control of Dengue-Malaria Co-infection with Local and Advanced Treatment Strategies
    (2026-01-01)
    Pongsumpun, Puntipa
    ;
    Ud Din, Rahim
    ;
    Ullah, Atta
    ;
    Pongsumpun, Puntani
    Abstract: This study presents a novel fractional-order co-infection model describing the joint transmission dynamics of dengue and malaria using the generalized fractional derivative. The total human population is divided into eight epidemiological compartments that account for single infections, co-infection, treatment stages, and recovery. The proposed framework incorporates memory effects and nonlocal behavior, offering a more realistic representation of disease progression compared to classical integer-order models. Local and advanced treatment strategies are introduced based on infection severity, allowing targeted intervention for both mild and co-infected cases. The fundamental mathematical properties of the model, including positivity, boundedness, existence, and uniqueness of solutions, are rigorously established. The basic reproduction number is derived, and both local and global stability of the disease-free equilibrium are analyzed using suitable Lyapunov functions. A statistical sensitivity analysis is performed to identify key parameters influencing disease transmission. Furthermore, optimal control strategies are formulated to minimize co-infection prevalence while reducing treatment and implementation costs. Numerical simulations validate the theoretical findings and demonstrate that fractional-order dynamics provide deeper insights into long-term disease behavior. The results offer valuable guidance for policymakers in designing effective and cost-efficient strategies to control dengue and malaria co-infection. Graphic Abstract: (Figure presented.) The