Finite difference method

Abstract: An analytical formulation is presented for the computation of scattering and transmission by general anisotropic stratified material. This method employs a first-order state-vector differential equation representation of Maxwell's equations whose solution is given in terms of a 4 \times 4 transition matrix relating the tangential field components at the input and output planes of the anisotropic region. The complete diffraction problem is solved by combining impedance boundary conditions at these interfaces with the transition matrix relationship. A numerical algorithm is described which solves the state-vector equation using finite differences. The validation of the resultant computer program is discussed along with example calculations.

Abstract: In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.

Abstract: In this paper the three‐dimensional finite difference method is used to simulate borehole wave propagations in an isotropic as well as an anisotropic formation. The finite difference results agree excellently with the analytic solutions of a point force source in the transversely isotropic medium. The finite difference synthetics are also in very good agreement with the discrete wave‐number solutions for fluid‐filled borehole wave propagation. The finite difference synthetics are compared with ultrasonic lab measurements in a scaled borehole model. The borehole is drilled along the X axis in an orthorhombic phenolite solid. Both monopole and dipole logs agree well. The observations of the shear wave splitting in the dipole logs are confirmed by the finite difference simulations. The 3‐D finite difference method is applied to the fluid‐filled borehole wave propagation in the tilted isotropic formation and in the orthorhombic phenolite formation. In a borehole drilled along the Z axis in a phenolite formati...

Abstract: The present analysis concentrates to examine the influence of both thermal and solutal stratification on magneto-hydrodynamics (MHD) nanofluid flow along an exponentially stretching sheet. Moreover, simultaneous effects of mixed convection and viscous dissipation are also analyzed to determine the thermal conductivity within the restricted domain. Energy and concentration equation consist of two important slip mechanisms, namely: the Brownian motion of nanoparticles and the thermophoresis due to concentration difference. By the mean of compatible similarity transformed, a system of PDEs is converted into the system of nonlinear ODEs. The resulting nonlinear ODEs are successfully solved via the implicit finite difference method (FDM). Obtained numerical solutions are plotted for each profile for different and converging values of including parameters. To validate the results, numerical values of Nusselt number are compared with the existing literature for a particular case. Obtained results present the significant impact of each parameter on temperature and concentration. Nanofluid flow behaviour is also observed via velocity profile.