The Fredholm-Volterra integral equation of the second kind with continuous kernels with respect to position and time, is solved numerically, using the Collocation and Galerkin methods. Also the error, in each case, is estimated.

The goal of this paper is study the mixed integral equation with singular kernel in two-dimensional adding to the time in the Volterra integral term numerically. We established the problem from the plane strain problem for the bounded layer medium composed of different materials that contains a crack on one of the interface. Also, the existence of a unique solution of the equation proved. Therefore, a numerical method is used to translate our problem to a system of two-dimensional Fredholm integral equations (STDFIEs). Then, Toeplitz matrix (TMM) and the Nystrom product methods (NPM) are used to solve the STDFIEs with Cauchy kernel. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the methods. The codes were written in Maple.

The two-dimensional magnetohydrodynamics incompressible flow of nanofluid about a stretching surface is investigated with the existence of viscous dissipation and Joule heating. Moreover, the impact of the convective condition and mass suction is applied with the viscous nanofluid containing copper nanoparticles and the base fluid water. The similarity variables have been employed to transform the coupled nonlinear partial differential equations into the ordinary differential equations and the numerical scheme bp4c is implemented for the further analysis of the solution. The diverse results of temperature, skin friction coefficient, velocity, and the Nusselt number according to numerous parameters have been shown graphically. It appears that the Nusselt number and the skin friction reduces, which is caused by the enhancement of both Hartman number and nanoparticles concentration. Moreover, the fluid temperature surges with the growth of Biot number, and Eckert number whereas the growth of nanoparticles concentration and suction parameter diminishes the velocity and temperature profile. The inclusion of a significant quantity of nanoparticles in the base fluid increases the density of the corresponding nanofluids and accordingly the temperature of the coupled nanoparticles in the base fluids can be modified. Hence, nanofluids build an outstanding performance in electronic components appliances and other electrical devices. The existing research is further effective in refrigerators for stabilizing their rate of cooling.

No abstract

No abstract