Bernoulli sub-ODE Method Research Articles

Exact solutions of Hirota–Maccari system forced by multiplicative noise in the Itô sense

In this article, the stochastic Hirota–Maccari system forced in the Itô sense by multiplicative noise is considered. We just use the He’s semi-inverse method, sine–cosine method, and Riccati–Bernoulli sub-ODE method to get new stochastic solutions which are hyperbolic, trigonometric, and rational. The major benefit of these three approaches is that they can be used to solve similar models. Furthermore, we plot 3D surfaces of analytical solutions obtained in this article by using MATLAB to illustrate the effect of multiplicative noise on the exact solution of the Hirota–Maccari method.

Two efficient methods for solving the generalized regularized long wave equation

ABSTRACT In this paper, an exact method named Riccati–Bernoulli sub-ODE method and a numerical method named Subdomain finite element method are proposed for solving the nonlinear generalized regularized long wave (GRLW) equation. For this purpose, Bäcklund transformation of the Riccati–Bernoulli equation and sextic B-spline functions are used for the exact and numerical solutions, respectively. The single soliton wave motion is used to confirm the methods which are found to be correct and effective. The three invariants (, and ) of motion have been assessed to indicate the conservation properties of the numerical algorithm. For the motion of single solitary and error norms are handled to evaluate differences between the analytical and numerical solutions. Unconditional stability is demonstrated using von-Neumann method. The procured outcomes show that our new schemes guarantee an evident and a functional mathematical equipment for solving nonlinear evolution equations.

Exact traveling waves for the Fisher’s equation with nonlinear diffusion

In this paper, the Fisher’s equation is studied with three different forms of nonlinear diffusion. When studying population problems, various forms of nonlinear diffusion can capture the effects of crowding or aggregation processes. Exact solutions for such nonlinear problems can be extremely useful to practitioners in the field. The Riccati–Bernoulli sub-ODE method is employed to obtain the exact traveling wave solutions for our nonlinear diffusion equation. The solutions that we find are new and to our knowledge, have not been reported in the literature.

Optical solitons with M-truncated derivative and conservation laws for NLSE equation which describe pseudospherical surfaces

The present work analyzes the optical solitons with M-truncated and conservation laws (Cls) of the nonlinear Schrodinger equation that explain pseudospherical surfaces. The well-known integration scheme which is generalized Bernoulli sub-ODE method is utilized to construct such optical soliton solutions. For the successful existence of the solutions, the constraints conditions have been presented. The discussion for the physical features of the obtained solutions is reported. Moreover, we establish Cls for the equation under consideration by means of multiplier approach.

Disturbance solutions for the long–short-wave interaction system using bi-random Riccati–Bernoulli sub-ODE method

This article applied the Riccati–Bernoulli (RB) sub-ODE method in order to get new exact solutions for the long–short-wave interaction (LS) equations. Namely, we obtain deterministic and random solutions, since we consider the proposed method in deterministic and random cases. The RB sub-ODE technique gives the travelling wave solutions in forms of hyperbolic, trigonometric and rational functions. It is shown that the proposed method gives a robust mathematical tool for solving nonlinear wave equations in applied science. Furthermore, some bi-random variables and some random distributions are used in random case corresponding to the LS system. The stability for the obtained solutions in random case is considered. In addition, there is a display of several numerical simulations, which helps to understand the physical phenomena of these soliton wave solutions.

New exact and numerical solutions with their stability for Ito integro-differential equation via Riccati–Bernoulli sub-ODE method

This paper points out several exact travelling wave solutions for -dimensional Ito integro-differential equation via the Riccati–Bernoulli sub-ODE approach. We also aim to develop a numerical solution of the respective equation using central finite difference formulas. The stability of the presented exact and numerical solutions is also deduced using the Hamiltonian system and Von Neumann's concept, respectively. Moreover, the numerical schemes are studied in terms of their accuracy. The relative error arising from executing the numerical method is exhibited. We compare our results with others published in some articles. The accomplished numerical solutions are successfully compared with the analytical ones. The used processes can be extended to solve more integrable problems as well as non-integrable ones.

Optical Solitons With M-Truncated and Beta Derivatives in Nonlinear Optics

This paper studies optical solitons with M-truncated and beta derivatives (BD) for the Complex Ginzburg-Landau equation (CGLE) with Kerr Law nonlinearity. Two well-known integration schemes which are generalized tanh method (GTM) and generalized Bernoulli sub-ODE method (GBM) are utilized to extract such optical soliton solutions. For the successful existence of the solutions, the constraints conditions have been presented. The discussion for the physical features of the obtained solutions is reported.

The new exact solutions for the deterministic and stochastic (2+1)-dimensional equations in natural sciences

ABSTRACTThis paper poses the Riccati–Bernoulli sub-ODE method in order to find the exact (random) travelling wave solutions for the (2+1)-dimensional cubic nonlinear Klein–Gordon (cKG) equation and the (2+1)-dimensional nonlinear Zakharov–Kuznetsov modified equal width (ZK-MEW) equation. The obtained travelling wave solutions are expressed by the hyperbolic, trigonometric and rational functions. Indeed, these solutions reflect some interesting physical interpretation for nonlinear phenomena. We discuss our method in deterministic case and in a random case. Additionally, we can show and discuss this method under some random distributions. Finally, some three-dimensional graphics of some solutions have been illustrated.

Dynamics of optical solitons incorporating Kerr dispersion and self-frequency shift

This paper deals with the dynamics of optical solitons in nonlinear Schrödinger equation (NLSE) with cubic-quintic law nonlinearity in the presence of self-frequency shift and self-steepening. This type of equation describes the ultralarge capacity transmission and traveling of laser light pulses in optical fibers. Two robust analytical approaches are employed to determine contemporary solutions. Some new explicit rational, periodic and combo periodic soliton solutions are extracted using the extended trial equation method. The Riccati–Bernoulli sub-ODE method provided us with singular and dark soliton solutions. The constraints found are necessary for the existence of solitons.

Symmetry analysis, exact solutions and numerical approximations for the space-time Carleman equation in nonlinear dynamical systems

This work presents an investigation and analysis for the space-time Carleman equation (STCE) in nonlinear dynamical systems. We compute the point symmetries, similarity variable, similarity transformation for STCE with Riemann-Liouville (RL) derivative and reduce STCE to ordinary differential equation (ODE) of fractional order. The exact solutions with conformable derivative are obtained via the generalized Bernoulli (GB) sub-ODE method. The well known residual power series technique (RPST) is used to compute the corresponding approximate solutions for the obtained exact solution. We then verify the convergence analysis and error estimate of RPST. Numerical simulations of the results are shown with the aid of graphical illustrations and tables. We prove that the RPST is very efficient for investigating the numerical approximations for a system of fractional differential equations.

New exact solutions to the dual-core optical fibers

In this article, new exact solutions of the dual-core optical fibers model are determined by using the Riccati–Bernoulli sub-ODE method. As a result, rational, trigonometric, hyperbolic and exponential functions and a variety of special solutions like kink-shaped, bell-type soliton solutions are established. We also consider the linear stability analysis for the dual-core optical fibers model. We also plot the 3D graphs for some selected solutions with the aid of the MATLAB software. We finally compare our results with the other methods and show that the Riccati–Bernoulli sub-ODE method is superior to other methods.

A Riccati–Bernoulli sub-ODE Method for Some Nonlinear Evolution Equations

Abstract This article concerns with the construction of the analytical traveling wave solutions for the model of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave and the higher-order nonlinear Schrödinger equation by Riccati–Bernoulli sub-ODE method. We give the exact solutions for these two equations. The proposed method is effective tool to solve many other nonlinear partial differential equations. Moreover, this method can give a new infinite sequence of solutions. These solutions are expressed by hyperbolic, trigonometric and rational functions. Finally, with the aid of Matlab release 15, some graphical simulations were designed to see the behavior of these solutions.

The Deterministic and Stochastic Solutions of the NLEEs in Mathematical Physics

This paper poses the new Riccati–Bernoulli Sub-ODE method in order to find the exact and random traveling wave solutions for the $$(1+1)$$ -dimensional nonlinear dispersive modified Benjamin–Bona and the Drinfel’d–Sokolov–Wilson equation. We use this method in deterministic case as well as a random case. Additionally, we can show and discuss this method under some random distributions.

Solitary wave solutions for some nonlinear time-fractional partial differential equations

In this work, we have considered the Riccati–Bernoulli sub-ODE method for obtaining the exact solutions of nonlinear fractional-order differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The space–time fractional modified equal width (mEW) equation and time-fractional generalised Hirota–Satsuma coupled Korteweg–de Vries (KdV) equations are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations (ODEs), which were obtained from the nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

Soliton structures to some time-fractional nonlinear differential equations with conformable derivative

This research presents new soliton structures to some time-fractional nonlinear differential equations (TFNDEs) with conformable derivative. The powerful Ricatti–Bernoulli (RB) sub-ODE method is used to carry out the soliton solutions. Some of the obtained solutions include trigonometric, periodic wave and hyperbolic solutions. The constraint conditions for the existence of solitons are also presented. The RB sub-ODE method proves to be efficient and effective for the extraction of soliton structures for different types of TFNDEs. Some interesting figures for the numerical simulation of the obtained results are presented.

Applications of IBSOM and ETEM for solving a discrete electrical lattice

This paper presents a number of new solutions obtained for solving the nonlinear electrical transmission lines by using the improved Bernoulli sub-ODE method and the extended trial equation method. The proposed solutions are kink soliton solution, hyperbolic solution, trigonometric solution and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and concise and can be also applied to other nonlinear partial differential equations. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to construct many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in telecommunication systems where solitons are used for codify data.

Optical solitons and conservation law of Kundu–Eckhaus equation

This paper obtains optical soliton solutions to the Kundu–Eckhaus equation with general coefficients. The Riccati–Bernoulli's sub-ODE method as well as Kudryashov's scheme are employed to obtain soliton solutions to the model. This generalized earlier reported result of the model with specific coefficients. Subsequently, the multiplier approach was utilized to secure a conserved quantity with the bright soliton solution that was reported earlier.

Applications of IBSOM and ETEM for solving the nonlinear chains of atoms with long-range interactions

This paper presents a number of new solutions obtained for solving a complex nonlinear equation describing dynamics of nonlinear chains of atoms via the improved Bernoulli sub-ODE method (IBSOM) and the extended trial equation method (ETEM). The proposed solutions are kink solitons, anti-kink solitons, soliton solutions, hyperbolic solutions, trigonometric solutions, and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and succinct and can be also applied to other nonlinear models. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to establish many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in Hamilton’s equations and generalized momentum where solitons are used for long-range interactions.

Solitons and conservation laws to the resonance nonlinear Shrödinger's equation with both spatio-temporal and inter-modal dispersions

The resonant nonlinear Shrödinger's equation (RNLSE) with both spatio-temporal (STD) and inter-modal (IMD) dispersions which describes the modelling of fluids and propagation dynamics of optical solitons is studied using three analytical schemes. These are generalized projective-Riccati equation method (GPRE), Bernoulli sub-ODE method and the Riccati-Bernoulli sub-ODE. The presented problem is studied with Kerr law nonlinearity. Dark optical, singular, and combined formal solitons are acquired. The constraint conditions that naturally fall out of the solution structure guarantee the existence of these solitons. We derive the Lie point symmetry generators of a system of partial differential equations (PDEs) obtained by decomposing the underlying equation into real and imaginary components. Then we used these symmetries to construct a set of nonlocal conservation laws (Cls) using the technique introduced by Ibragimov.

New solitary wave solutions and conservation laws to the Kudryashov–Sinelshchikov equation

In this manuscript, we utilized the algorithm of Riccati–Bernoulli sub-ODE method to find new soliton solutions to the Kudryashov–Sinelshchikov equation (KSE). Some new type of traveling wave solutions are acquired, which includes the kink-type, singular-type and exponential function solutions which have not been obtained in previous time using this technique. The obtained solutions appear with all necessary constraint conditions that are necessary for them to exist. Using the general conservation laws (Cls) theorem introduced by Ibragimov, the Cls for the underlying equation are investigated.