The extended Power Series (XPS) method can be extremely useful for solving nonlinear equations with regular and irregular singular points. The extended power Series is considered times a logarithm or times a fractional power of x, etc.). This research shows it is simple to solve approximately the Perkins-Kern-Nordgren (PKN) model of hydraulic fracture. To illustrate the effectiveness and convenience of the XPS method, we consider the two cases of dimensionless PKN equation containing the M-scaling and -scaling. The results compared with available analytical results verified excellent agreements.

Orpiment, getchellite and stibnite from Zarshuran deposit, NW Iran, exhibit extensive As-Sb substitution. There are four known minerals along As2S3-Sb2S3 join, i.e. orpiment (As2S3), Wakabayashilite [(As,Sb)20S30], getchellite (AsSbS3) and stibnite (Sb2S3). Extensive mineralogical studies by scanning electron microscope (SEM) and electron probe micro-analyzer (EPMA) indicate that getchellite from Zarshuran exhibits large As/Sb substitution, compared to the synthetic mineral and occurrences in other deposits, which justifies further experimental studies. Natural orpiment with the highest Sb content was recorded in Zarshuran samples. Considering the possible explanation for the incomplete record of solid solution, it appears that rapid variation in fluid chemistry and variability of As and Sb concentration in the hydrothermal solution may have caused extensive solid solution in the As2S3-Sb2S3 system. The recorded solid solution may indicate that temperatures of gold mineralization were less than 250oC which is supported by fluid inclusion data.

A general initial-boundary value problem of one-dimensional transient wave propagation in a multi-layered elastic medium due to arbitrary boundary or interface excitations (either prescribed tractions or displacements) is considered. Laplace transformation technique is utilised and the Laplace transform inversion is facilitated via an unconventional method, where the expansion of complex-valued functions in the Laplace domain in the form of general Dirichlet Series is used. The final solutions are presented in the form of finite Series involving forward and backward travelling wave functions of the d’ Alembert type for a finite time interval. This elegant method of Laplace transform inversion used for the special class of problems at hand eliminates the need for finding singularities of the complex-valued functions in the Laplace domain and it does not need utilising the tedious calculations of the more conventional methods which use complex integration on the Bromwich contour and the techniques of residue calculus. Justification for the solutions is then considered. Some illustrations of the exact solutions as time-histories of stress or displacement of different points in the medium due to excitations of arbitrary form or of impulsive nature are presented to further investigate and interpret the mathematical solutions. It is shown via illustrations that the one-dimensional wave motions in multi-layered elastic media are generally of complicated forms and are affected significantly by the changes in the geometrical and mechanical properties of the layers as well as the nature of the excitation functions. The method presented here can readily be extended for three-dimensional problems. It is also particularly useful in seismology and earthquake engineering since the exact time-histories of response in a multi-layered medium due to arbitrary excitations can be obtained as finite sums.

In this paper, we use Taylor Series and Legendre functions of the second kind to remove singularity of the weakly singular Fredholm integral equations of the second kind with the kernel k (x, y) =1/(x-y)a 0<a<1. Legendre polynomials are used as a basis and some integrals that appear in this method are computed with Cauchy principal value sense without using any numerical quadrature. Three examples are given to show the efficiency of the method.