Numerical solutions of one-dimensional heat and advection-diffusion equations are obtained by collocation method based on cubicB-spline. Usual finite difference scheme is used for time and space integrations. CubicB-spline is applied as interpolation function. The stability analysis of the scheme is examined by the Von Neumann approach. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.

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B-splines have been widely used to approximate solutions to differential equations. In this paper, a class of singular boundary value problems are treated by using extended cubic uniform B-spline approximations. The advantage of using an extended cubic B-spline rather the ordinary B-spline is that it introduces one additional free parameter. For a number of examples where exact solutions are known, the solutions obtained using the extended B-splines are found to be better approximations than those obtained using ordinary B-splines.

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A computational model is developed to analyze the unsteady flow of blood through stenosed tapered narrow arteries, treating blood as a two-fluid model with the suspension of all the erythrocytes in the core region as Herschel-Bulkley fluid and the plasma in the peripheral layer as Newtonian fluid. The finite difference method is employed to solve the resulting system of nonlinear partial differential equations. The effects of stenosis height, peripheral layer thickness, yield stress, viscosity ratio, angle of tapering and power law index on the velocity, wall shear stress, flow rate and the longitudinal impedance are analyzed. It is found that the velocity and flow rate increase with the increase of the peripheral layer thickness and decrease with the increase of the angle of tapering and depth of the stenosis. It is observed that the flow rate decreases nonlinearly with the increase of the viscosity ratio and yield stress. The estimates of the increase in the longitudinal impedance to flow are considerably lower for the two-fluid Herschel-Bulkley model compared with those of the single-fluid Herschel-Bulkley model. Hence, it is concluded that the presence of the peripheral layer helps in the functioning of the diseased arterial system.