Al-Azhar University, Faculty of Science (Boys)Al-Azhar Bulletin of Science1110-253531Issue 1-B20200601RADIATION EFFECT ON MAGNETOHYDRODYNAMIC CARREAU NANOFLUID FLOW PAST NON-LINEAR PERMEABLE HEATED STRETCHING SHEET WITH PARTIAL SLIP11311148010.21608/absb.2020.111480ENHameda M.ShawkyDepartment of Mathematics, Faculty of Science, Al-Azhar University.Nabil T.EldabeDepartment of Mathematics, Faculty of Education, Ain Shams University.Kawther A.KamelDepartment of Mathematics, Faculty of Science, Al-Azhar University.Esmat A.Abd-AzizDepartment of Mathematics, Faculty of Science, Al-Azhar University.Journal Article20200115The motion of a Carreau nanofluid past an infinite vertical non-linear permeable stretching sheet embedded in a non-Darcy porous medium is investigated. It is stressed by a uniform external magnetic field. The thermal radiation and heat generation are taken in consideration as well as Brownian motion with thermophoresis, chemical reaction and partial slip velocity at the boundary layer. The nonlinear partial differential equations describing the motion with heat and mass transfer are transformed to non-linear ordinary differential equations and solved by using Runge-Kutta method with approbriate boundary conditions. The effects of the physical parameters of the problem on the obtained solutions are discussed numerically and graphically. Through the section of discussion the skin friction and the rate of heat and mass transfer are computed. It is found that these physical parameters play an important rules to control the obtained velocity, temperature and concentration of the fluid. https://absb.journals.ekb.eg/article_111480_d4bba146088ad14edc3fb43b8ebfaa73.pdfAl-Azhar University, Faculty of Science (Boys)Al-Azhar Bulletin of Science1110-253531Issue 1-B20200601STABILITY ANALYSIS IN THE RESTRICTED FOUR BODY PROBLEM WITH OBLATNESS AND RADIATION PRESSURE.122211148310.21608/absb.2020.111483ENM. N.IsmailAstonomy and Meteorology Department, Faculty of Science, Al-Azhar University, EgyptSahar H.YounisMath. Dep., Faculty of Science (Girls), Al-Azhar University, EgyptGhada F.MohamdienNational Research Institute of Astronomy and Geophysics (NRIAG), EgyptJournal Article20200112In the present work, the canonical form of the differential equations is derived from the Hamiltonian function H which is obtained for the system of the four-body problem. This canonical form is considered as the equations of motion, the equilibrium points of the restricted four-body problem are studied under the effects of radiation pressure and oblatness Lyapunov function is used to provide a method for showing that equilibrium points are stable or asymptotically stable. If the system has an equilibrium point conditionally the eigenvalues of the system contain negative real parts, the scalar potential function is positive definite, then The Lyapunov center’s theorem is used to analyze the stability and periodicity of the motion of orbits about these equilibrium points of the restricted four-body problem. From this theorem, the Lyapunov function is found. Also, the stability regions are studied by using The Poincare maps, an analytical and numerical approach had been used. A cod of Mathematica is constructed to truncate these steps. The periodic orbits around the equilibrium points are investigated for the Sun-Earth-Moon system.https://absb.journals.ekb.eg/article_111483_ee0ea50ccc62a53d8eceac76517263e1.pdfAl-Azhar University, Faculty of Science (Boys)Al-Azhar Bulletin of Science1110-253531Issue 1-B20200601A NEW IMPROVED SCHEME FOR PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS IN SEMI-INFINITE DOMAIN VIA DOUBLE RATIONAL CHEBYSHEV FUNCTIONS223511148410.21608/absb.2020.111484ENMahmoud Abd El GhanyNassarAl Azhar university mathematics departmentKamal R.RaslanDepartment of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, EgyptMohamed RamadanFaculty of Science, Menoufia University, Shebein El-Koom, Egypt.Journal Article20200204The concept of double rational Chebyshev functions on the semi-infinite domain () and some of their properties are introduced in this work. Also, the definition of derivatives for double rational Chebyshev functions is improved. This new definition is employed to deal with partial differential equations with variable coefficients derived on the interval. The new definition with the spectral collocation method generates a new improved scheme. Numerical results are show that demonstrates the validity and applicability of the two techniques. The obtained numerical results are compared with the exact solution where it shown to be very attractive with good accuracy.https://absb.journals.ekb.eg/article_111484_681606661586744dad9c850a01d0757f.pdf