In Mitsui (Sci Eng Rev Doshisha Univ Jpn 51(3):181–190, 2010) the author introduced a new class of discrete variable methods known as look-ahead linear multistep methods (LALMMs) which consist of a pair of predictor-corrector (PC), including a function value at one more step beyond the present step for the numerical solution of ordinary differential equations. Two-step family of Look-Ahead linear multistep methods of fourth-order pair were derived and shown to be A(\(\theta \))-stable in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181–188, 2011). The derived integration methods are of low orders and unfortunately cannot cope with stiff systems of ordinary differential equations. In this paper we extend the concept adopted in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181–188, 2011) to construct second-derivative of high-order accuracy methods with off-step points which behave essentially like one-step methods. The resulting integration methods are A-stable, convergent, with large regions of absolute stability, suitable for stiff systems of ordinary differential equations. Numerical comparisons of the new methods have been made and enormous gains in efficiency are achieved.

In this research work, we investigate the influence of heat source and chemical reaction on electro-magneto-hydrodynamic (EMHD) blood flow through bifurcated arteries with external tilted magnetic field for treating tumor. The potential electric field applied along the bifurcated arterial wall, accurately described the Poisson-Boltzmann equation. We modeled the EMHD blood flow to obtain the non-dimensionalized form of the equations. We converted the modeled equations to ordinary differential equations by the use of suitable variables. Exact solutions of the converted equations are calculated by the method of undetermined coefficients and the results obtained with the aid of Mathcad software were simulated and presented graphically. From the graphical representation of results, we observed that increase in Joule heating and Eckert number, increases the temperature distribution in the affected tumor cells which prevent high thermal radiation exposure from killing the healthy cells within the tumor region. The curves of the wall shear stress seem to be greater in the converging region in comparison to the diverging region, but when the strength of magnetic field and thermal radiation parameters increase the wall shear stress decreases at the bifurcated wall where blockage may likely occur due to the development of boundary layers on the inner walls of the bifurcated region. By increasing the heat radiation parameter, we observed that the curves representing both velocity and temperature profiles increase rapidly from the origin and the velocity of blood flow varies directly for the converging, diverging and the tumor regions of the bifurcated arteries. Thus, thermal radiation effect is a result of the higher rate of heat transfer at the vessel walls. Further, combining the electromagnetic field and the heat radiation together gives new insights of the physical properties of blood flow in the body system network which is essential for health practitioners and clinicians.

In this paper, we construct transient electro-osmotic flow of Burgers’ fluid with Caputo fractional derivative in a micro-channel, where the Poisson–Boltzmann equation described the potential electric field applied along the length of the microchannel. The analytical solution for the component of the velocity profile was obtained, first by applying the Laplace transform combined with the classical method of partial differential equations and, second by applying Laplace transform combined with the finite Fourier sine transform. The exact solution for the component of the temperature was obtained by applying Laplace transform and finite Fourier sine transform. Further, due to the complexity of the derived models of the governing equations for both velocity and temperature, the inverse Laplace transform was obtained with the aid of numerical inversion formula based on Stehfest's algorithms with the help of MATHCAD software. The graphical representations showing the effects of the time, retardation time, electro-kinetic width, and fractional parameters on the velocity of the fluid flow and the effects of time and fractional parameters on the temperature distribution in the micro-channel were presented and analyzed. The results show that the applied electric field, electro-osmotic force, electro-kinetic width, and relaxation time play a vital role on the velocity distribution in the micro-channel. The fractional parameters can be used to regulate both the velocity and temperature in the micro-channel. The study could be used in the design of various biomedical lab-on-chip devices, which could be useful for biomedical diagnosis and analysis.

The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta methods suitable for solving stiff differential systems. The lack of such stability properties makes the continuous solution not reliable, especially in solving large stiff differential systems. We derive these methods by using one intermediate off-grid point in between the familiar grid points for continuous solution within the interval of integration. The new family had a high accuracy, non-overlapping piecewise continuous solution with very low error constants and converged under the suitable conditions of stability and consistency. The results of computational experiments are presented to demonstrate the efficiency and usefulness of the methods, which also indicate that the block hybrid methods are competitive with some strong stability stiff integrators.

In this paper, we report the effects of fractional relaxation time on the parameters of blood flow together with magnetic particles through straight circular cylindrical arterial segment. A mathematical model of blood flow subject to pulsatile pressure gradient in the axial direction with external magnetic field applied normal to the direction of flow is presented. Combining the momentum equation together with the Maxwell model parameter appropriately, leads to the governing fractional partial differential equation which permits to obtain the velocity profile of blood along with magnetic particles. By adopting the non-dimensionalized form of the new version of the governing fractional partial differential equation allowed us to obtain the dimensionless relaxation time parameter λ₁ which controls blood flow conditions. Solving the fractional partial differential equations using Laplace and finite Hankel transforms we found that the influence of the order of Caputo's fractional time-derivative and fractional relaxation time on the blood flow parameters with magnetic particles are enormous. The graphical results plotted of different influential parameters are presented and discussed in details. The velocities of blood flow and that of magnetic particles are reduced under the influence of the external magnetic field and the relaxation time parameter. The magnetic particles are assumed to be uniformly distributed within the blood, since they are flowing in the same axial direction designated by along a circular cylindrical coordinates of radius. This is a very good indication that blood velocity can be controlled by the application of external magnetic field as well as the relaxation time parameter during treatment to avoid tissues damage. The present study has important applications in magnetic field control of biotechnological processes, bio magnetic device technology, biomedical engineering and pathology. Keywords: Arterial segment, Blood flow, Relaxation time, Magnetic field, Magnetic particles