Simple Equation Method Research Articles

Staility-Indcatng RP-HPLC Method Development and Validation for Simulaneous Estimaton of Gliclazde and Chromium Picolinate in Bulk and Formulaton

Simultaneous determination of Gliclazide and Chromium Picolinate two component methods has been developed in methanol. In this method, the overlapping spectra of Gliclazide and Chromium Picolinate were well resolved by making use of the zero order spectra of their direct absorption spectra. The method was based on the measurement of absorbance of Gliclazide and Chromium Picolinate at 272.00nm, 262.00 nm and 272.05nm respectively. This method obeyed Beer’s law in the concentration range of 0 to 120 μg/mL for Gliclazide and Chromium Picolinate. The method was validated as per the ICH guideline and accuracy, precision are found to be within the acceptable limit. The limits of detection and quantitation were found to be 0.023 and 0.070 μg/ml, respectively for Gliclazide, and 0.021 and 0.063 μg/ml, respectively for Chromium Picolinate . A simple, specific, accurate and precise Simultaneous Equation Method for three Components Spectrophotometric methods was developed and validated for simultaneous estimation of Gliclazide and Chromium Picolinate in synthetic mixture. This method is validated as per ICH Q2R1Guidelines.

Bright solitons of the model with arbitrary refractive index and unrestricted dispersion

Objective:The generalized Schrödinger equation with arbitrary refractive index and unrestricted order is considered. The objective of this paper is to demonstrate the bright solitons of this mathematical model can be found by the same formula taking into account the order of nonlinear differential equation. Method:The algorithm of constructing optical solitons of the generalized nonlinear Schrödinger equation with unrestricted dispersion using the simplest equation method is presented and applied for finding of optical solitons. Results:The optical solitons of the generalized nonlinear Schrödinger equation of the twelfth order are found at three powers on nonlinearity. The application of the method for finding solution of the generalized Korteweg–de Vries equation with arbitrary power of nonlinearity and unrestricted dispersion is discussed.

A new perspective for analytical and numerical soliton solutions of the Kaup–Kupershmidt and Ito equations

In present study, it is aimed to implement the analytical method and a numerical approach based on finite element method to get approximate soliton solutions of two special forms of fifth-order KdV equation (fKdV) that are of particular importance: Kaup–Kupershmidt (K–K) and Ito equations. To reach these aims, the simplest equation method and septic B-spline collocation method are introduced. L2 and L∞ error norms are computed for single soliton solutions to demonstrate the proficiency and accuracy of the present method. The method is shown to be unconditionally stable by performing the von-Neumann stability analysis. The obtained numerical results have been displayed with tables. Additionally, to show the analytical and numerical behaviors of the single soliton, all figures are drawn in 2D and 3D. Analytical and numerical results ensure that the methods are more suitable and systematically handle the solution procedures of nonlinear equations arising in mathematical physics.

Assorted hyperbolic and trigonometric function solutions of fractional equations with conformable derivative in shallow water

We present the modified simple equation method for solving nonlinear fractional-order partial differential equations (fPDEs). With the presented method, some important fPDEs are solved. We used conformable derivatives to solve these equations. The solutions obtained are both hyperbolic and trigonometric function solutions. Most of these solutions are new solutions not found in the literature.

A new computational investigation to the new exact solutions of (3 + 1)-dimensional WKdV equations via two novel procedures arising in shallow water magnetohydrodynamics

Abstract Various new exact solutions to ( 3 + 1 ) \left(3+1) -dimensional Wazwaz–KdV equations are obtained in this work via two techniques: the modified Kudryashov procedure and modified simple equation method. The 3D plots, contour plots, and 2D plots of some obtained solutions are provided to describe the dynamic characteristics of the obtained solutions. Our employed techniques are very helpful in constructing new exact solutions to several nonlinear models encountered in ocean scientific phenomena arising in stratified flows, shallow water, plasma physics, and internal waves.

Some exact solutions of the Yu–Toda–Sasa–Fukuyama equation with time-dependent coefficients via two different methods

In the current study, the generalized (G′/G)-expansion method and a new modified simple equation method (NMSEM) have been utilized to secure the exact solutions of the Yu-Toda-Sasa-Fukuyama (YTSF) equation with time-dependent coefficients, which describes two layers of liquid (or in an elastic) media. By implementing these methods, we have secured the three different families of travelling wave solutions (such as hyperbolic, rational and trigonometric functions) of the independent variables and the arbitrary parameters of the equation. Further, the graphical representation of the solitons are done with the help of the software Maple.

Conservation Laws and Travelling Wave Solutions for a Negative-Order KdV-CBS Equation in 3+1 Dimensions

In this paper, we study a new negative-order KdV-CBS equation in (3+1) dimensions which is a combination of the Korteweg-de Vries (KdV) equation and Calogero–Bogoyavlenskii–Schiff (CBS) equation. Firstly, we determine the Lie point symmetries of the equation and conservation laws by using the multiplier method. The conservation laws will be used to obtain a triple reduction to a second order ordinary differential equation (ODE), which lead to line travelling waves and soliton solutions. Such solitons are obtained via the modified form of simple equation method and are displayed through three-dimensional plots at specific parameter values to lend physical meaning to nonlinear phenomena. It illustrates that these solutions might be extremely beneficial in understanding physical phenomena in a variety of applied mathematics areas.

The analysis of solitonic, supernonlinear, periodic, quasiperiodic, bifurcation and chaotic patterns of perturbed Gerdjikov–Ivanov model with full nonlinearity

In this manuscript, the perturbed Gerdjikov–Ivanov equation is considered for the investigating model. It is studied to govern the dynamics of soliton propagation through optical fibers, metamaterials or PCF. The observed model is subjected to an extended simple equation technique, which disclosed an abundant of exact solutions including trigonometric function, singular, kink, peakon and compacton solutions. These solutions are made available with their essential conditions, which guarantee the persistence of such optical solitons and portraits using appropriate physical parameters in 3D, 2D and density plots. After that, the traveling wave transformation is used to turn the mth-order nonlinear perturbed Gerdjikov–Ivanov equation into a planar dynamical system. The dynamical and chaotic behaviors of the considered equation are also discussed. The qualitative analysis of the dynamical system and the chaotic behaviors of the perturbed system are investigated using the theory of plane dynamic systems. In addition, we utilize the RK4 method to discover patterns in the dynamical system that seem to be super nonlinear, periodic, and quasiperiodic. The reported results are new and have not been investigated. They can be used in the explanation of the physical phenomena modeled and will give information about the long-term dynamic behavior. Numerical simulations reveal that by changing the frequency and amplitude parameters have an impact on the dynamic behaviors of the system. It is established that the extended simple equation method and dynamical observations offer further influential mathematical tools for constructing exact solutions and their qualitative analysis in NLEEs in mathematical physics.

Wave solutions of the DMBBM equation and the cKG equation using the simple equation method

In this article, we transform the (1 + 1)-dimensional non-linear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and the (2 + 1)-dimensional cubic Klein Gordon (cKG) equation, which are the non-linear partial differential equations, into the non-linear ordinary differential equations by using the traveling wave transformation and solve these solutions with the simple equation method (SEM) with the Bernoulli equation. Two classes of exact explicit solutions-hyperbolic and trigonometric solutions of the associated NLEEs are characterized with some free parameters; we obtain the kink waves and periodic waves.

Development and Validation of UV Spectrophotometric Methods for Simultaneous Estimation of Dolutegravir Sodium and Rilpivirine Hydrochloride in Pure Bulk Form

A simple, specific, sensitive, rapid, precise, accurate and economical UV Spectrophotometric simultaneous equation method have been developed and validated for the routine estimation of Dolutegravir sodium and Rilpivirine Hydrochloride in bulk form. The Method A employs estimation of drugs by simultaneous equation method (SEM) using synthetic mixture of drugs. The absorption maxima of drugs were found to be at 259.20nm for Dolutegravir sodium and 305.80nm for Rilpivirine Hydrochloride. Both drugs followed the beer lamberts law in the range of 4-24µg/ml and 1-8µg/ml for DOL and RIL respectively. Methods are validated according to ICH guidelines and can be adopted for the routine analysis Dolutegravir sodium and Rilpivirine Hydrochloride in pure bulk form.

New unexpected soliton solutions to the generalized (2 + 1) Schrödinger equation with its four-mixing waves

The propagation of solitons in birefringent fiber is one of the phenomena that has an important role in all modern technological means of communication. The generalized [Formula: see text] nonlinear Schrödinger equation (GNLSE) with its four-mixing waves (FMW) is the famous model that describes the propagation of solitons in birefringent fiber perfectly. In fact, the FMW governed effectively the performance of the resultant solitons amplitude. Hereby, we will study this model to extract new unexpected optical soliton solutions to this model via various three techniques. The three famous methods that are a candidate for this purpose are the extended direct algebraic method (EDAM), the extended simple equation method (ESEM) and the solitary wave ansatz method (SWAM). The three techniques are implemented successively for the suggested model to establish the optical solutions of the suggested model successfully. The optical soliton solutions that are achieved by these proposed techniques give surprise expectations that weren’t achieved previously by any other authors who used other techniques.

New impressive representations for the soliton behaviors arising from the (2+1)-Boussinesq equation

In the present work, we will detect unpredicted conducts “which weren't realized before” to the soliton solutions of the (2+1)-Boussinesq equation by using three distinct algorithms. The proposed equation is the famous nonlinear one which distinguishes the waves of coastal and ocean engineering that involve the nonlinearity and dispersion terms. Moreover, it is a developed form of the standard Boussinesq equation which describes the solution interaction mechanism of shallow-water waves that involve many waves and shallow water influences refraction, diffraction, shoaling and weak nonlinearity properties arising in fluid dynamics. It also plays principal role in many physics branches, such as propagation of long waves in shallow water, vibrations in a nonlinear string and ion sound waves in plasma and one-dimensional nonlinear lattice waves. The three algorithms that will select for this purpose are the (G′G) -expansion method, the extended direct algebraic method (EDAM) and the extended simple equation method (ESEM). These three distinct manners are applied in the same time and parallel. We will show comparison between our new soliton solution behaviours with that constructed before.

Spatio-temporal dynamic solitary wave solutions and diffusion effects to the nonlinear diffusive predator-prey system and the diffusion-reaction equations

The coupled parabolic nonlinear evolution equations that describe the temporal-spatio dynamics of predator-prey systems and the nonlinear diffusion-reaction equations with diffusion effects have broad applications in biology, population dynamics, ecology, and many other fields where nonlinearity and diffusivity are key features of wave evolution. As a result, exact traveling wave solutions of such models are highly effective in numerical and analytical theories. In this article, we find the soliton solutions of the above stated biological models by contriving the enhanced modified simple equation method which is related to the exponential, hyperbolic, and trigonometric functions by balancing the highest order nonlinearity and diffusivity associated with biological groups. Graphical representations of some of the obtained solutions in three- and two-dimensional layouts are provided to estimate their behavior. The results revealed that the stated method is categorical, robust, and efficient in finding exact solutions to a variety of nonlinear evolution equations.

Wave Effects of the Fractional Shallow Water Equation and the Fractional Optical Fiber Equation

The non-linear space-time fractional Estevez-Mansfield-Clarkson (EMC) equation and the non-linear space-time fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation showed the motion of waves in the shallow water equation and the optical fiber equation, respectively. The process used to solve these equations is to transform the non-linear fractional partial differential equations (PDEs) into the non-linear ordinary differential equations by using the Jumarie's Riemann-Liouville derivative and setting the solution in the finite series combined with the simple equation (SE) method with the Bernoulli equation. The new traveling wave solutions were the exponential functions resulting in the physical wave effects are produced in the form of kink waves and represented by the two-dimensional graph, three-dimensional graph, and contour graph. In addition, the comparison of the solutions revealed that the new solutions have a more convenient and easier format.

Optimal system of Lie subalgebra for symmetry reductions, group invariant solutions and exact solutions to the coupled Hirota–Maccari system driving pulse propagation in optical fiber

In this paper, Lie symmetry analysis has been proposed by utilizing the Lie group of continuous point transformation for obtaining the new exact soliton solutions of the (2+1)-dimensional Hirota–Maccari system. Lie infinitesimals and possible geometric vector fields are obtained by applying the third-order prolongation on this system. Also, their commutative product and adjoint relations have been presented in Tables 1 and 2. By considering the resulting symmetries, one-dimensional optimal system of Lie subalgebra is obtained. Meanwhile, the Hirota–Maccari system is reduced to a system of ordinary differential equations with the help of optimal subalgebras. Furthermore, the simplest equation method has been used to obtain the abundant exact soliton solutions of the reduced system. At last, conservation laws of Hirota–Maccari system have been extracted by utilizing the generalized “new conservation theorem” invoked by Ibragimov.

Optical solitons in polarization preserving fibers for perturbed resonant NLSE with Kerr law nonlinearity and Bohm potential having multiplicative white noise via Itô calculus

Object:We discuss for the first time, the perturbed resonant nonlinear Schrödinger equation (NLSE) with Kerr law nonlinearity and Bohm potential having a multiplicative white noise. Methods:Three integration schemes via modified simple equation method, sine–cosine method and Jacobi elliptic equation method are used. Results:We retrieve soliton solutions to the model, which are described using parametric constraints. Jacobi elliptic function solutions, bright, dark, singular soliton solutions are presented.

Novel Optical Soliton Waves in Metamaterials with Parabolic Law of Nonlinearity via the IEFM and ISEM

Here, the miscellaneous soliton solutions of the generalized nonlinear Schrödinger equation are considered that describe the model of few-cycle pulse propagation in metamaterials with parabolic law of nonlinearity. The novel analytical wave solutions to the mentioned nonlinear equation in the sense of the nonlinear ordinary differential transform equation are obtained. The techniques are the improved exp − Γ ϖ function method and the improved simple equation method. The nonlinear ordinary transform to concern the generalized Schrodinger equation to convert it for a solvable integer-order differential equation is used. After the successful implementation of the presented methods, the exact solitary wave solutions in the form of trigonometric, rational, and hyperbolic functions are obtained. Hence, the presented methods are relatable and efficient to solve nonlinear problems in mathematical physics.

Solitary Wave Solutions for Generalized Boiti–Leon–Manna–Pempinelli Equation by Using Improved Simple Equation Method

Solitary Wave Solutions for Generalized Boiti–Leon–Manna–Pempinelli Equation by Using Improved Simple Equation Method

Lump, interaction of lump and kink and solitonic solution of nonlinear evolution equation which describe incompressible viscoelastic Kelvin–Voigt fluid

In this script, we consider the modified Oskolkov equation in incompressible viscoelastic Kelvin–Voigt fluid and fluid dynamics. A dominant direct algebraic method namely modified simple equation method (MSE) uses to retrieve various dynamical structural solutions of the nonlinear models. As a results, we can derive a disguise version of analytic structural solutions in exponential, hyperbolic, trigonometric and Jacobi elliptic functions with some appreciate parameters. In this work, we depict the physical explanation of cross periodic lump wave, interaction of lump and kink, lump and anti-kink shape solution, double periodic wave, kink, anti-kink shape solution, disguise version of soliton solutions with 3D contour plot and 2D plot.

Abundant analytical optical soliton solutions to the integrable (3+1)-Boussinesq equations

In our current work, we will extract abundant analytical optical soliton solutions to the (3+1)-Boussinesq equation ((3+1)-BE) by using three different techniques that are implemented for the first time to this model. The suggested model is the developed one that governs the coastal and ocean engineering waves that involve the nonlinearities and the dispersion terms as the tsunami waves and other versions of solid mechanics waves. The three suggested techniques that will be choosed to extract these unexpected abundant analytical optical soliton solutions to the (3+1)-Boussinesq equation are the extended direct algebraic method (EDAM), the (G′G) -expansion method and the extended simple equation method (ESEM) which are implemented in arranged sequence and with the same vein. Moreover, the novelty of our obtained solutions clear when the comparison is implemented not only between our new extracted abundant analytical optical soliton solutions that are demonstrated by these three methods but also with that established previously by other authors who used other various methods.