The main contribution of this paper is to prove the existence of extremal solutions for a novel class of $\mathsf{\psi }$-Caputo fractional differential equation with nonlinear boundary conditions. For this purpose, we utilize the well-known monotone iterative technique together with the method of upper and lower solutions. Finally, we provide an example along with graphical representations to confirm the validity of our main results.

In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.

The numerical study of natural convection heat transfer inside a U-shaped enclosure with various thermal profiles by using copper–alumina/water hybrid nanofluid is reported. The study focuses on the effect of constant, linear, quadratic, and sinusoidal with different amplitudes and periods and absolute sinusoidal thermal profiles to the heat transfer efficiency inside the lid. A weighted residual Galerkin triangular finite element method is used alongside the damped Newton–Raphson algorithm to calculate the heat transfer rate of the system in terms of averaged and local Nusselt number. The result from the present model is compared with the previously published experimental and numerical data. The isotherms and streamlines of the fluid flow for each thermal profile are presented with a nanoparticle volume fraction of between 0 and 0.1 for Rayleigh number of 10⁴–10⁶. It is found that the constant heating profile gives the best heat dissipation performance, while the sinusoidal thermal profile performed the worst. The results gained from this paper will be useful in understanding how heat can be transferred effectively as it can be applied to many different industries, including the electronics field.