Our main focus in this paper is to investigate the effects of Soret and Dufour known as thermodiffusion and diffusion-thermo on moving plate in copper water nanofluid. The set of partial differential equations are converted into set of ordinary differential equations using the appropriate similarity variables before being solved numerically using bvp4c code in Matlab software. The results of heat and mass transfer, temperature and concentration profiles on Soret as well as Dufour effects are presented graphically. Soret effect increases the heat transfer rate at the surface while Dufour effect decreases the mass transfer rate at the surface. Since the solutions exist in dual, we carry out the stability solutions to determine which solution is stable and hence the physical meaning is realized physically.

The investigation concerning the movement of the viscous fluid and heat transfer rate when the cylinder shrinks is carried out. The role of curvature parameter and effect of cylinder parameter on skin friction and heat transfer rate were studied. The existence of a dual solution is observed and subsequently identifies which solution is stable. By using the similarity transformation, the boundary layer equations are transformed into nonlinear ordinary differential equations which are then solved numerically using bvp4c in MATLAB. The results for the skin friction coefficient, temperature gradient coefficient as well as velocity and temperature profiles are presented graphically. The effects of mass suction parameter and curvature parameter on flow and heat transfer characteristics are also presented. It is found that the dual solution existed when mass suction parameter greater or equal to 2. Cylinder surface diminished the performance of skin friction coefficient and heat transfer rate. The stability analysis result is performed to verify that the upper branch solution is stable and physically meaningful.

An investigation is conducted to study the flow and heat transfer on stagnation point over an exponentially stretching/shrinking cylinder filled with nanofluid in the presence of slip at the boundary. By using the appropriate exponential similarity transformation, the governing equations are converted into nonlinear ordinary differential equations and then solved computationally using bvp4c in Matlab software. The results of skin friction coefficient, heat transfer coefficient, velocity and temperature profiles on slip parameter, curvature parameter, nanoparticles as well as nanoparticle volume fraction parameter are presented graphically. The presence of slip and curvature parameters cause the region of dual solutions to expand and at once enhance the heat transfer rate at the surface but somehow the heat transfer rate at the surface decreases rapidly when cylinder is shrunk. The aim of this paper is to investigate the effect of slip parameter on nanofluid as well as on the stretching/shrinking surface. The new findings of the effects of skin friction and heat transfer coefficient on different nanoparticles and nanoparticle volume fraction were discussed. Since there are dual solutions in the flow characteristics, we carry out a stability analysis to verify which solution is in a stable state and can be realized physically

The current study proposes a numerical method which solves nonlinear Fredholm and Volterra integral of the second kind using a combination of a Newton–Kantorovich and Haar wavelet. Error analysis for the Holder classes was established to ensure convergence of the Haar wavelets. Numerical examples will illustrate the accuracy and simplicity of Newton–Kantorovich and Haar wavelets. Numerical results of the current method were then compared with previous well-established methods.