On the exact analytical solution of some families of equilibrium critical thickness transcendental equations

Abstract

The problem of finding an exact analytical closed-form solution of some families of transcendental equations, which describe the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films, is studied in some detail by the Special Trans Functions Theory (STFT). A novel STFT mathematical approach with an analytical closed-form solution is presented. Structure of the STFT exact solutions, numerical results and graphical simulations confirm the validity of the basic principle of the STFT. The proposed STFT analytical approach shows qualitative improvement in theoretical sense (a novel gradient coefficient genesis), and, in accuracy when compared to the conventional analytical and numerical methods.

Concerning an analytical solution of some families of Kepler’s transcendental equation

“…5 polynomialization of Kepler's equation through Chebyshev polynomial expansion of the sine is presented. , Hopfield neuron analysis 14 some families of transcendental equations, 8,12,15,20,[23][24][25]38 solar cell analysis, 16,21,35 Plutonium temperature estimation, 8,17,36 ambient temperature estimation, 18,11 Lambert transcendental equations analysis, 8,20 as well as in problem in engineering materials, 22,34,40,41 ect. The biggest impact of this work is in the usage of a new STF theory approach in formulae genesis for the determining position (r, v) in time t. All investigations and analysis of the value E in this paper have been consistently accomplished with the usage of STF theory.…”

“…[8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]34,35 The Kepler's transcendental equation for determining the eccentric anomaly in celestial mechanics (1) after some symbolic modification of the type…”

Determining the STFT exact formulae to the transistor thermal sensitivity estimations

“…Let us note that Perovich's Special Trans Functions Theory (STFT) has been proved to be a very powerful theory for solving the broad families of transcendental equations, some integral and integral -differential equations, and obtaining exact analytical closedform solutions. Examples of the STFT application are shown in papers concerning the determining of the closed-form solutions in: theory of neutron slowing down [1][2][3], nonlinear circuit theory [1], [4], [18][19], linear transport theory [1], [5][6], [15], Hopfield neuron analysis [7], some families of transcendental equations [1], [8], solar cell analysis [9], [13], [17], [21], Plutonium temperature estimation [1], [10], ambient temperature estimation [1], [11], Lambert transcendental equations analysis [1], [12], as well as in problem in engineering materials [14], [16], etc.…”

The Special Trans Functions Theory for the Degree of Nuclear Fuel Burn-Up Estimation

“…(3)) is solvable using the Special Tran Function Theory [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Let us note that Perovich's Special Tran Functions Theory (STFT), has been proved to be a very powerful theory for solving transcendental equations, some integral and integro-differential equations, and, obtaining exact analytical closed-form solutions.…”

“…Let us note that Perovich's Special Tran Functions Theory (STFT), has been proved to be a very powerful theory for solving transcendental equations, some integral and integro-differential equations, and, obtaining exact analytical closed-form solutions. Examples of STFT application are shown in articles concerning the closed-form solution genesis in: the theory of neutron slowing down [ [6], [7], [16]], the nonlinear circuit theory [ [8], [16], [22]], the linear transport theory [ [9], [16], [20]], the Hopfield neuron analysis [11], some families of transcendental equations analysis [ [12], [16], [17]], the solar cell theoretical analysis [ [13], [18], [22]], the Plutonium temperature estimation [14], the ambient temperature estimation [ [15], [16]], the Lambert transcendental equations analysis [ [16], [17], [23]], as well as in the problem in the engineering materials [19], [21], [22]], etc.…”