Riccati Nonlinear Differential Equation Research Articles

Analytical solutions via coupled Elzaki adomian decomposition method for some applications

An efficient combination of Adomian Decomposition iterative technique coupled Elzaki transformation (ETADM) for solving Telegraph equation and Riccati non-linear differential equation (RNDE) is introduced in a novel way to get an accurate analytical solution. An elegant combination of the Elzaki transform, the series expansion method, and the Adomian polynomial. The suggested method will convert differential equations into iterative algebraic equations, thus reducing processing and analytical work. The technique solves the problem of calculating the Adomian polynomials. The method’s efficiency was investigated using some numerical instances, and the findings demonstrate that it is easier to use than many other numerical procedures. It has also been established that (ETADM) is a trustworthy technique for solving differential equations. Using the Mathematica 13.3 programme, the graphs of the approximate solutions are presented.

Modelling J-Curve by Using Differential Equations: Studies in Finances and Entrepreneurship

In this paper, a mathematical formalism for defining the J-Curve phenomenon is set up in the form of a corresponding differential equation with practical application. An explicit form of the differential equation to describe the J-Curve, as well as the solution of the differential equations in terms of polynomials with coefficients satisfying a particular property was presented for the first time. The J-Curve and S-Curve are modelled as Riccati's nonlinear differential equation of the 1^st order. A mathematical form of the differential equation is proposed that corresponds in structure to the Laguerre Polynomials (linear differential equation of the 2^nd order), resulting in J-Curve as a solution. To confirm this, it is necessary to fulfil two main criteria for the mathematical validation of the J-Curve - confirmation of the structure of Laguerre polynomials (a polynomial equation with coefficients of alternating sign) and an R-squared score greater than 0.6. Two case studies are presented to validate the theoretical concepts and demonstrate the application of J-Curve mathematical modelling - Returns on Venture Investments (where returns from start-up investments are analysed over a 12-year period) and Returns on Long-Term Stock Investments (based on actual financial data for the period from 2016 to 2020 for the NIFTY 500 Index of the National Stock Exchange of India). A direct connection between the mathematical formulation and the graphs obtained is shown, which corresponds to the mathematical validation of the J-Curve phenomenon. It is mathematically shown how the financial data manifest the J-Curve behaviour, satisfying the initial assumptions of such a model. The mathematical model set up in this way can be verified in practice on different other types of data, which could create interest in an interdisciplinary approach in such research. This could include studies particularly from other economic fields such as micro- or macroeconomics.

The Harmonic Oscillator in the Framework of Scale Relativity

The dynamical law obeyed by the one-dimensional physical systems in the scale relativity approach is reduced to a Riccati nonlinear differential equation. Applied to the harmonic oscillator potential, we show that such an approach permits the calculation of the solutions of the scale relativity problem in terms of the well known solutions of the Schrödinger equation for the harmonic oscillator.

Exact solution of the Schrödinger equation with a new expansion of anharmonic potential with the use of the supersymmetric quantum mechanics and factorization method

The study involves finding exact eigenvalues of the radial Schrodinger equation for new expansion of the anharmonic potential energy function. All analytical calculations employ the mathematical formalism of the supersymmetric quantum mechanics. The novelty of this study is underlined by the fact that for the first time the recurrence formulas for rovibrational bound energy levels have been derived employing factorization method and algebraic approach. The ground state and the excited states have been determined by means of the hierarchy of the isospectral Hamiltonians. The Riccati nonlinear differential equation with superpotentials has been solved analytically. It has been shown that exact solutions exist when the potential and superpotential parameters satisfy certain supersymmetric constraints. The results obtained can be utilized both in computations of quantum chemistry and theoretical spectroscopy of diatomic molecules.

Riccati equation‐based generalization of Dawson's integral function

AbstractA new generalization of Dawson's integral function based on the link between a Riccati nonlinear differential equation and a second‐order ordinary differential equation is reported. The MacLaurin expansion of this generalized function is built up and to this end an explicit formula for a generic cofactor of a triangular matrix is deduced. Copyright © 2007 John Wiley & Sons, Ltd.

Transient Optical Regime of Two Level Atom in Laser Light

An extended analytic approach is considered for optical Bloch equations in the two level atom interacting with laser light. The separation approach of coupled differential equations is always possible with a sequence of special transformation into the Riccati nonlinear differential equation. The conditions that permit an exact solutions of three coupled system are extracted in a natural manner. The case of sodium atom moving along the axis of a monochromatic wave is treated in some details including a discussion on the radiation pressure forces exerted by laser light in the transient regime.

An analytic approach to a three-level atom interacting with a single-mode quantized field

An analytic approach is proposed for three-level systems, based on the Riccati nonlinear differential equation. We extract in a natural manner a set of conditions that permit an exact solution for a three-level atom interacting with a single-mode field. We obtain closed analytic exact solutions for the probability amplitudes and apply this approach to the Jaynes–Cummings model of three-level atoms.

Iterative Step-Like Reconstruction of Stratified Dielectric Media from Multifrequency Reflected-Field Data

We describe in this paper a frequency domain microwave method to reconstruct the permittivity profile of layered dielectric half-space. The inversion algorithm is based on the iterative Newton–Kantorovitch procedure applied to the Riccati nonlinear differential equation. To obtain a correction term to the initial profile, the integral equation relating to a small change of the profile function and corresponding change of the reflection coefficient is solved twice, with different systems of expanding functions for the unknown profile. The first set of functions is utilized to determine positions of the reflecting interfaces between layers. The second set represented by the Heaviside step functions provides the profile correction term completing the iteration. The proposed approach is valid for the essentially band-limited input data, that is, when both low- and high-frequency information is not available. This alleviates practical implementation especially from the viewpoint of antenna design, dielectric rods fed by TEM horns are used as transmitting/receiving antennas. Some numerical and experimental examples are presented for the input data in the frequency range of 1–4GHz, which demonstrate validity and robustness of the method for the complicated highly contrasted dielectric profiles.

A DIRECT METHOD OF OPERATIONAL CALCULUS (I): —Research for the solution of second-order linear ordinary differential equations with variable coefficients

According to Mikusinski's fundamental view, we reconstruct a direct method of operation a calbulus which may be adapted to search for the solution of differential equations with variable coefficients.The present paper is devoted to solutions of two types—indirect (undetermined) method and direct (series of integral) method-for solving second-order linear differential equations: The latter displays forthright feature while the former can solve second-order linear differential equation and interrelate first-order Riccati's non-linear differential equation at the same time.

Riccati's Nonlinear Differential Equation

(1960). Riccati's Nonlinear Differential Equation. The American Mathematical Monthly: Vol. 67, No. 2, pp. 134-139.

Riccati's Nonlinear Differential Equation

Riccati's Nonlinear Differential Equation