In this paper, Touchard polynomials (TPs) are presented for solving Linear Volterra integral equations of the second kind (LVIEs-2k) and the first kind (LVIEs-1k) besides, the singular kernel type of this equation. Illustrative examples show the efficiency of the presented method, and the approximate numerical (AN) solutions are compared with one another method in some examples. All calculations and graphs are performed by program MATLAB2018b.

A new method based on the Touchard polynomials (TPs) was presented for the numerical solution of the linear Fredholm integro-differential equation (FIDE) of the first order and second kind with condition. The derivative and integration of the (TPs) were simply obtained. The convergence analysis of the presented method was given and the applicability was proved by some numerical examples. The results obtained in this method are compared with other known results.

Background: Synthetic hydroxyapatite,(Ca10(PO4)6(OH2) can directly bond to bones without infection and fibrous encapsulation, thus is regarded as bioactive and biocompatible. The aim of the study was the estimation of microarchitecture bone parameters include bone mass (gm/cm2) cortical bone width (mm), thread width (mm), marrow space star volume analysis (V*m) and osteoblast, osteocyte cell number. Materials and methods: Ninety-six (96) commercially pure titanium CpTi) used in this study, (48) implants were coated with HA by dipping coating and (48) implants were used as control. They were inserted in (32) Newzland white rabbits and followed for 2 & 6 weeks. Mechanical torque removal test and histomorphometric analysis of bone microarchit

In this research, deposition of titanium oxide (TiO₂) and vanadium oxide (V₂O₅) thin film  in different mixing percentage  (0, 25 ,50, 75 and100)%   on the substrate of glass .The coating thickness was ( 50 nm  ).

In this research contact angle was measured and the effect of weather conditions. Results showed that the value of the contact angle of the prepared films reached its highest value at 50% (TiO₂+V₂O₅) was 160º.

The results showed that the optical transmittance of TiO₂ and V₂O₅ thin film decrease with increasing the deposition angle and decrease with increasing V₂O₅ pro

The real and imaginary part of complex dielectric constant for InAs(001) by adsorption of oxsagen atoms has been calculated, using numerical analysis method (non-linear least square fitting). As a result a mathematical model built-up and the final result show a fairly good agreement with other genuine published works.

TiO2 thin films have been deposited at different concentration of
CdO of (x= 0.0, 0.05, 0.1, 0.15 and 0.2) Wt. % onto glass substrates
by pulsed laser deposition technique (PLD) using Nd-YAG laser
with λ=1064nm, energy=800mJ and number of shots=500. The
thickness of the film was 200nm. The films were annealed to
different annealing (423 and 523) k. The effect of annealing
temperatures and concentration of CdO on the structural and
photoluminescence (PL) properties were investigated. X-ray
diffraction (XRD) results reveals that the deposited TiO2(1-x)CdOx
thin films were polycrystalline with tetragonal structure and many
peaks were appeared at (110), (101), (111) and (211) planes with
preferred orientatio

Asmari is the main productive reservoir in Abu Ghirab oilfield in the south-east part of Iraq. It has history production extends from 1976 up to now with several close periods. Recently, the reservoir suffers some problems in production, which are abstracted as water production rising with oil production declining in most wells. The water problem type of the field and wells is identified by using Chan's diagnostic plots (water oil ratio (WOR) and derivative water oil ratio (WOR') against time). The analytical results show that water problem is caused by the channeling due to high permeability zones, high water saturation zones, and faults or fracturing. The numerical approach is also used to study the water movement inside the reser

Asmari is the main productive reservoir in Abu Ghirab oilfield in the south-east part of Iraq. It has history production extends from 1976 up to now with several close periods. Recently, the reservoir suffers some problems in production, which are abstracted as water production rising with oil production declining in most wells. The water problem type of the field and wells is identified by using Chan's diagnostic plots (water oil ratio (WOR) and derivative water oil ratio (WOR') against time). The analytical results show that water problem is caused by the channeling due to high permeability zones, high water saturation zones, and faults or fracturing. The numerical approach is also used to study the water movement insi