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    Exact analytical wave solutions for space-time variable-order fractional modified equal width equation
    (Elsevier, 2022) Ali, Umair; Ahmad, Hijaz; Baili, Jamel; Botmart, Thongchai; Aldahlan, Maha A.
    The variable-order evolution equation is an impressive mathematical model that explain complex dynamical problems efficiently and accurately. This latest research investigates the modified equal width nonlinear space–time variable-order fractional differential equation and fractional derivative operator in the sense of Caputo. Using transformation, an ordinary differential equation is obtained from the variable-order differential equation. For accuracy, the space–time variable-order fractional modified equal width equation is solved by the modified (G’/G) -expansion method. This model describes the simulation for one-dimensional wave transmission in nonlinear media with dispersion processes. As a result, new traveling wave solutions are developed for various values of parameters. The obtained solutions have several applications in a recent area of research in mathe- matical physics. The obtained graphical solutions are in the form of singular solitons, kink solitons and periodic solitons waves which demonstrate that the physical signifance and dynamical behaviors of fractional variable- order differential equations and the proposed method are more effective, powerful, and easy in solving prob- lems arising in mathematical physics
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    Fractional mathematical modeling of malaria disease with treatment & insecticides
    (Elsevier, 2022) Sinan, Muhammad; Ahmad, Hijaz; Ahmad, Zubair; Baili, Jamel; Murtaza, Saqib; Aiyashi M.A.; Botmart, Thongchai
    Many fatal diseases spread through vertical transmission while some of them spread through horizontal transmission and others transmit through both modes of transmission. Horizontal transmission illnesses are usually carried by a vector, which might be an animal, a bird, or an insect. Plasmodium parasites that dwell in red blood cells produce malaria, an infectious illness. This parasite is mostly transmitted to humans via mosquitoes. The dynamics of Malaria illness among human persons and vectors are examined in this study. The impact of the vector (mosquito) on disease transmission is also taken into account. The problem is described using nonlinear ODEs that are then generalized using the Atangana–Baleanu fractional derivative. Some theoretical analyses such as existence and uniqueness and stability via Ulam–Hyres stability analysis and optimal control strategies have been done. The numerical solution has been achieved via a numerical technique by implementing MATLAB software. Results of fractional, as well as classical order, are portrayed through different graphs while some figures are displayed for the global asymptotical stability of the model. From the graphical results, it can be noticed that the control parameters drastically decrease the number of infected human and vector population which will off course minimize the spread of infection among the human population. In addition to that, from the graphical results, it also be noticed that our model is globally asymptomatically stable as the solution converges to its equilibrium. Moreover, the use of bednets and insecticides can reduce the spread of infection dramatically while the impact of medication and treatment on the control of infection is comparatively less.