Abstract: Power total number of primes from the discharge of the decimal system is identified by the law of exponential growth with 14 fundamental physical constants. Model obtained on the parameters of the physical constants, proved less of the error and it gives more accurate predictions of the relative power of the set of prime numbers. The maximum absolute error of power (the number of primes), the traditional number is three times higher than suggested by us complete a number of prime numbers. Therefore, the traditional number 2, 3, 5, 7,. .. is only a special case. The transformation In10=2,30285… it was a rough rounded, leading to false identification of physico-mathematical regularities of different series of prime numbers. Model derived from physical constants, proved more accurate than the relative accuracy, and it gives more accurate predictions of the relative power of the set of prime numbers with increasing discharge the decimal number system.

Abstract: In this paper, a matrix method based on collocation points on any interval [a,b] is proposed for the approximate solution of some second order nonlinear ordinary differential equations with the mixed conditions in terms of Fermat polynomials. The method, by means of collocation points, transforms the differential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Fermat coefficients. Also, the method can be used for solving Riccati equation. The numerical results show the effectiveness of the method for this type of equation. Comparing the methodology with some known techniques shows that the present approach is relatively easy and high accurate.

Abstract: In this paper, we present a new approximate method for solving systems of nonlinear Fredholm integral equation. This method is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Numerical results and comparisons with the exact solution are included to demostrate the validity and applicability of the technique.

Abstract: In this paper substantiated for a differential equation of pseudo-parabolic type with discontinuous coefficients a contact-boundary value problem with non-classical boundary conditions is considered, which requires no matching conditions. Equivalence of these conditions boundary condition is substantiated classical, in the case if the solution of the problem in the izotropic S.L. Sobolev's space is found. The considered equation as a pseudo-parabolic equation generalizes not only classic equations of mathematical physics (heat-conductivity equations, string vibration equation) and also many models differential equations (Aller's equation, Manjeron equation, telegraph equation, moisture transfer generalized equation, Boussinesq - Love equation and etc.). It is grounded that the contact-boundary conditions in the classic and non-classic treatment are equivalent to each other, and such boundary conditions are demonstrated in geometric form. Even from geometric interpretation can see that the grounded non-classic treatment doesn't require any additional conditions of agreement type. Thus, namely in this paper, the non-classic problem with contact-boundary conditions is grounded for a pseudo-parabolic equation. For simplicity, this was demonstrated for one model case in one of S.L. Sobolev izotropic space W_p^(4,4)(G).

Abstract: In this work, the code ‘Laue’ was used to simulate the diffraction pattern and to investigate the effects of varying the golden mean in a monatomic linear quasicrystal having a pseudo atomic potential. The work involved setting the parameters of the code required to simulate the diffraction, running the code and exporting the data generated to excel for analysis. It was found that the shape of the diffraction pattern and the background intensity for a given value of the golden mean is unique. Both the width of the diffraction pattern and the intensity of the central peak decrease with increasing golden mean. Results obtained illustrated the features of the diffraction by quasicrystal and proved the suitability and accuracy of the code in simulating the dynamics of quasicrystals.