A numerical method for special Cosserat rods based on Antman's description Antman, 2005 is developed for hyperelastic materials and potential forces. This method preserves the relevant properties of the underlying PDE system, namely, the orthonormality of the directors and the conservation of the energy.

The problem considered is an investigation of the possible collapse of the roof between the pillar next to be mined in secondary coal mining and the first line of pillar remnants called snooks. The roof rock between the pillar, which is the working face, and the snook is modelled as an Euler-Bernoulli beam acted on at each end by a horizontal force and by its weight per unit length. The beam is clamped at the pillar and simply supported (hinged) at the snook. The dimensionless differential equation for the beam and the boundary conditions depend on one dimensionless number $B$. We consider the range of values of $B$ before the displacement and curvature first become singular at $B=B_1$. The model predicts that for all practical purposes, the beam will break at the clamped end at the pillar. The failure of the beam for values of $B$ greater than $B_1$ is investigated computationally.

Fractional variational iteration method (FVIM) is performed to give an approximate analytical solution of nonlinear fractional Riccati differential equation. Fractional derivatives are described in the Riemann-Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. The behavior of the solutions and the effects of different values of fractional order $\alpha$ are indicated graphically. The results obtained by the FVIM reveal that the method is very reliable, convenient, and effective method for nonlinear differential equations with modified Riemann-Liouville derivative

We employ the Heat Balance Integral Method (HBIM) to solve a number of thermal and phase change problems which occur in a multitude of industrial contexts. As part of our analysis, we propose a new error measure for the HBIM that combines the least-squares error with a boundary immobilisation method. We describe how to determine this error for three basic thermal problems and show how it can be used to determine an optimal heat balance formulation. We then show how the HBIM, with the new error measure, may be used to approximate the solution of an aircraft deicing problem. Finally we apply the new method to two industrially important phase change problems.

Since 1960's, the blow-up phenomena for the Fujita type parabolic equation have been investigated by many researchers. In this survey paper, we discuss various results on the life span of positive solutions for several superlinear parabolic problems. In the last section, we introduce a recent result by the author.

Understanding and quantifying more of the workings of the human bladder motivates the present industry-supported study. The bladder performance in terms of the urinary velocities produced tends to be dominated by the internal fluid dynamics involved, in the sense that the bladder wall moves in a body-prescribed way. The enclosed urine flow responds to this wall movement, and there is relatively little feedback on the wall movement. Combined computational work and special-configuration analysis are applied over a range of configurations including computational and analytical results for the circle and sphere as basic cases; models of more realistic bladder shapes; the end stage of the micturition process where the bladder is relatively squashed down near the urethral sphincter and localised peak speeds arise. The combination of approaches above can be extended to allow for interaction between wall shape and flow properties such as internal pressure if necessary.