Transaction in Theoretical and Mathematical Physics
Volume:1 Issue: 1, Winter 2024

Throughout this paper, we apply the Optimal Homotopy Asymptotic Method (OHAM) to find out the numerical solutions of the fractional Cahn-Hilliard (C-H) equation. We examine fractional order time dependent partial differential equations to assess the method's competency. In the Caputo sense, fractional-order derivatives have been applied with numerical values in the closed interval [0, 1]. The most advantage of this method is that it contains parameters that strongly control the solution series convergence. Additionally, this method greatly simplifies calculations because it does not require any linearization, discretization, or little perturbations. Approximate solutions of the C-H equation were compared with the exact solutions; moreover the results of the suggested method have been compared with those of other widely used numerical techniques, such as the Adomian decomposition analysis method. A comparison of these solutions with the exact solution shows that our method is more effective and accurate for solving nonlinear differential equations. MATLAB R2021b is utilized to generate the numerical results.

Investigating complex systems and accessing information has been of interest to researchers for years. The desired system in this research includes the investigation of the plasma resulting from the interaction of the laser with the desired substance of the research. Air is the material of choice for producing plasma in interac-tion with the laser. We used Nd:YAG laser for incident beam to interact with air matter. By the interaction of laser with air, plasma was formed. In the following, it is possible to check the plasma density by scanning the plasma resulting from the interaction of the laser with air. Therefore, in order to determine the electron density by the Saha method, we first need to calculate the electron temperature, which was calculated using the spectral line pair intensity ratio method of the plasma temperature equal to Kelvin. In the continuation of the research, the electron density was measured using Saha's equation

In this paper, guiding of Gaussian laser pulse in plasma channel is numerically investigated. We assumed that the plasma channel has radial and longitudinal changes. We obtained the matched condition for guiding of high-intensity laser pulses through the plasma channel. Using the source-dependent expansion (SDE) method, we ex-tracted four paired equations for pulse amplitude, phase, spot size and inverse of the radius of wave front curvature. Numerical results of equations were obtained using the Runge-Kutta numerical method. Numerical results showed that the normalized laser spot size during propagation in the plasma channel is constant in the matched mode and oscillates for the mismatched mode. The Runge-Kutta method of order 4 has good results for well posed prob-lems. We have considered the functional form of the model and in this work we have shown that our problem is well posed, so the proposed method has satisfactory nu-merical results. The numerical results confirm this concept.

  Hydrogen atom is always helpful in obtaining datafrom the other atoms. Revision of the Lie algebraic approachin studying the radial part of the Hydrogen atom and usingthe Tilting transformation are reasonable means to discussAlkali elements. The characteristic difference between Alkalielements and Hydrogen atom is the screening effect inAlkali elements which causes much more freedom to thehighest electron, valence electron. In this case, these atomshave another terms besides coulomb potential energy; Sentencesof the order of inverse square of the distance andabove. So, we can consider we have dipole moments, quadrupolemoments and etc. Therefore, these terms , although theyare small, make these elements attractive for applications inquantum information. Also, we can consider them as Rydbergatoms with strong long-range interactions. Here, byconsidering an inverse square distance potential, Lie algebraicmethods help us to obtain energy levels of both boundand excited states and corresponding wave functions too.

  In this work, we have solved the radial Schrö-dinger equation for the Woods–Saxon potential together with coulomb (r>Rc), centrifugal terms and spin orbit in-teraction by using a new type of Nikiforov-Uvarov (NU) method. This approach is based on the solution of the Second-Order Linear Differential Equations (SOLDE). The mandatory specific choices of the required parameters in this technique restricts the application of this method to the Schrödinger equation with complicated potential pro-files, which means that the NU method cannot efficiently be employed to solve more realistic physical systems. Due to the mentioned difficulties in evaluating the equivalent second order algebraic equation in the NU method, the analytical NU method has to be extended to more effi-cient version that is combined with numerical methods (that leads to a semi-analytical method). We have solved it by combination of the NU method with the numerical fitting schema. The numerical fitting schema helps us to find the mentioned second order algebraic equation. Oth-erwise, complicated changes of variables or overwhelming algebraic treatments to deriving the energy eigenvalues and the wavefunctions are required. The current approach is simpler, more flexible and efficient. This technique can also be developed to be suitable for the equations other than the Schrodinger one. The Woods–Saxon potential is also a short-range interaction in the potential model for nuclear physics and has predictions for the nuclear shell model and distribution of nuclear densities. We have ob-tained a semi-analytical energy eigenvalues and eigen-functions for various values of n, l, and j quantum num-bers. Agreement of 5/2+ and 1/2+ wavefunctions with the published works is also obtained which also shows the ac-curacy of our method.

  In this paper we have obtained the adjoint of an arbitrary operator (linear and nonlinear) in Hilbert space by introducing n-dimensional Riemannian manifold. This general formalism covers every linear operator (non – dif-ferential) in Hilbert space. In fact, our approach shows that instead of using directly the adjoint definition of an operator, it can be obtained directly by relying on a suitable generalized space according to the action of the operator in question. For the case of nonlinear operators we have to change the definition of the linear operator adjoint. But here, we have obtained adjoint of these operators with respect to the definition of the derivative of the operator. In matter fact, we have shown one of the straight applications of ''Frechet derivative'' into algebra of the operators.1 IntroductionThis paper consists of two main parts. In first part, we look for a general relationship for the adjoint of the linear operators. In fact, we intend to achieve a universal formula for the adjoint of the linear operators with a generalized space such as Riemannian manifold. Although the self-adjoint extensions of differential operators on Riemannian manifold has been studied [1] but our main focus is on non - differential operators. Recently, it has been reported for linear operators to be unitary in curved space [2]In sec