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The fundamental intention of this article is to find several soliton solutions to the generalized Benjamin–Bona–Mahony equation. The fractional derivatives are described in the conformable sense. These solutions are generated using the functional variable approach. Then, we acquire via this technique some distinctive solutions, such as the singular soliton, periodic soliton, bell-shaped soliton, and kink soliton solutions. The fundamental feature of the demonstrated method is that it's convenient and provides precise, effective solutions to nonlinear equations. The solutions given will offer a thorough examination of this model and any associated occurrences. [ABSTRACT FROM AUTHOR]

In this paper, the technique involving the e x p a function is employed to calculate analytical soliton solutions for three distinct equations: the (3+1)-dimensional mKdV–Zakharov–Kuznetsov equation, the KdV equation, and the (1+1)-dimensional Mikhailov Novikov–Wang integrable equation, which fractional-order in the sense of conformable derivatives. By selecting some parameter values, a diverse spectrum of soliton solutions is obtained, encompassing kink solitons, singular solitons, and periodic-singular solitons. The representation in physical terms enables the examination of authentic multispecies plasmas, plasma models, and frequency ranges. [ABSTRACT FROM AUTHOR]

The current manuscript proposes an innovative approach for obtaining exact solutions to the conformable time-fractional nonlinear differential-difference equations (NDDEs). The fundamental concept of this approach involves the generalized exponential rational function method (GERFM). In this method, the exact solutions include trigonometric and hyperbolic functions. In order to assess this method's efficacy, we consider its application to the conformable time-fractional Hybrid Lattice equation. After solving this model, we provide its dynamic behavior through the 3D, 2D, and Contour graphs. The soliton solutions obtained are used to describe any related physical phenomena. [ABSTRACT FROM AUTHOR]

In this article, we focus on a specific version of the NDDEs which is the relativistic Toda lattice equation. We employ the generalized exponential rational function method on a nonlinear model of surface wave propagation to recognize their diverse singular soliton and multi-soliton wave structures. What is remarkable in this article is the use of graphic diagrams, which have diversified the solutions for solving such equations, leading to a greater understanding of the movements of particles and the strengthening of nonlinear lattice dynamics. The efficiency and strength of the employed method are illustrated, signifying its applicability to a wide spectrum of NDDEs in physical phenomena. [ABSTRACT FROM AUTHOR]

This article aims to study the generalized exponential rational function method to solve the differential-difference Burgers' equation. Our approach is an efficient method for solving nonlinear partial differential equations, and it can be used for a specific type of nonlinear differential-difference equations. To recognize diverse singular soliton and multi-soliton wave structures, we displayed the 3-D and contour graphs associated with the solutions. [ABSTRACT FROM AUTHOR]

The increasing capabilities of control systems, driven by abundant data and computational resources, necessitate the incorporation of data‐based techniques with model‐based approaches. While mathematical models exist for air and space vehicles, they may not fully meet the requirements for expanding control system capabilities. Consequently, the development of data‐based extensions in collaboration with these models becomes crucial to provide specific analysis and synthesis tools in the hybrid realm for practical cases. This article introduces the data‐assisted control (DAC) framework for aerospace vehicles, with the NASA generic transport model serving as the platform for study. By leveraging data, the model‐based controller enhances performance during a damage event. Real‐time decisions are made to update the control law based on information obtained from the data, while the model‐based controller may not exhibit consistent performance. The closed‐loop system is shown to be stable in the transition phase between the data and the model, and robust against uncertainties. The dual unscented Kalman filter is employed to estimate fixed dynamic parameters and generalized force‐moment, and the Koopman estimator is used to estimate the evolution equation of the estimated forces and moments. The dual estimation model ensures observability by considering the persistency of excitation. Simulations have shown that the purely model‐based robust control leads to degradation of the closed‐loop performance in case of damage, suggesting the need for data assistance. The DAC framework offers a robust approach to integrate data and models, significantly enhancing control performance in practical aerospace applications. [ABSTRACT FROM AUTHOR]

The principal purpose of this research is to utilize the exp-function approach for achieving exact solutions of nonlinear optical fibers, including fractional order in the sense of conformable derivatives. Owing to the algorithm of symbolic computational, Some soliton solutions are obtained, including solitary soliton, singular kink-type soliton, and periodic solitons. These results are generated and developed by using the exp-function. To the greatest of our knowledge, the soliton solutions given in this study could be highly beneficial in comprehending physical phenomena. [ABSTRACT FROM AUTHOR]

The major purpose of this study is to seek diverse soliton solutions to the nonlinear discretized mKdv lattice system including fractional-order in the sense of conformable derivative. The rational forms of trigonometric, hyperbolic, and Jacobi elliptic functions are employed to generate these exact solutions. The auxiliary rational structure is used for obtaining exact solutions which provide solitary, periodic, and soliton varieties of traveling wave solutions. The solutions obtained will provide a comprehensive study of this model and any related events. [ABSTRACT FROM AUTHOR]

The ultimate focus of this research is to provide soliton solutions to the nonlinear differential-difference equations via conformable fractional derivatives which arise from the polarization of the ferroelectric nanoparticles. The investigation of achieved soliton solutions will provide effective results of ferroelectric materials and their related phenomena. [ABSTRACT FROM AUTHOR]

The objective of the present study was to employ a green synthesis method to produce a sustainable ZnFe 12 O 19 /BiOI nanocomposite and evaluate its efficacy in the photocatalytic degradation of metronidazole (MNZ) from aqueous media. An artificial neural network (ANN) model was developed to predict the performance of the photocatalytic degradation process using experimental data. More importantly, sensitivity analysis was conducted to explore the relationship between MNZ degradation and various experimental parameters. The elimination of MNZ was assessed under different operational parameters, including pH, contaminant concentration, nanocomposite dosage, and retention time. The outcomes exhibited high a desirability performance of the ANN model with a coefficient correlation (R2) of 0.99. Under optimized circumstances, the MNZ elimination efficiency, as well as the reduction in chemical oxygen demand (COD) and total organic carbon (TOC), reached 92.71%, 70.23%, and 55.08%, respectively. The catalyst showed the ability to be regenerated 8 times with only a slight decrease in its photocatalytic activity. Furthermore, the experimental data obtained demonstrated a good agreement with the predictions of the ANN model. As a result, this study fabricated the ZnFe 12 O 19 /BiOI nanocomposite, which gave potential implication value in the effective decontamination of pharmaceutical compounds. [Display omitted] • Synthesized and characterized ZnFe 12 O 19 /BiOI as a new green magnetic nanocomposite. •The ZnFe 12 O 19 /BiOI/Xenon process was evaluated in actual pharmaceutical wastewater. •OH radicals was the prevailing species for photodecomposition of metronidazole. •The ZnFe 12 O 19 /BiOI was an efficient and highly stable photocatalyst. • Artificial neural network confirmed the laboratory data in this work. [ABSTRACT FROM AUTHOR]

In this paper, the generalized rational function procedure was used to produce new solitary solutions to the fractional Jaulent–Miodek hierarchy equation. The equation is shortly called the Jaulent–Miodek equation, which was first derived by Jaulent and Miodek and associated with energy-dependent Schrödinger potentials. The equation is converted into a fourth-order partial differential equation using a transformation. In order, we find some solitary solutions such as soliton, rational and hyperbolic function solutions of Jaulent–Miodek hierarchy equation by the help of generalized exponential rational function procedure. Then, for some parameters, we draw three-dimensional graphics of real values of some exact solutions that we acquired by using this procedure. [ABSTRACT FROM AUTHOR]

In this paper, we considered the nonlinear Schrodinger equation modeling cubic optical solitons in a polarization-preserving fiber with Kerr law. Using the improved t a n (φ / 2) -expansion method, a powerful and effective method, we constructed exact solutions including the hyperbolic function solution, the trigonometric function solution, the exponential solution and the rational solution with free parameters. The geometrical shapes for some of the obtained results are depicted for various choices of the free parameters that appear in the results. The obtained solutions are entirely new and can be considered as generalization of the existing results in the ordinary derivative case. [ABSTRACT FROM AUTHOR]

This paper studies the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili-Boussinesq (KPB) equation via the Hirota's bilinear form and symbolic computation. Mixed type lump solutions are presented, which include rational function, trigonometric function and hyperbolic function. The propagation and the dynamical behaviors of these mixed type of lump solutions are shown by some three-dimensional and contour plots. [ABSTRACT FROM AUTHOR]

This work concerns the solution of non-symmetric, sparse linear systems with multiple right hand sides by iterative methods. Herein a global version of the range restricted generalized minimal residual method (RRGMRES) is proposed for solving this sort of problems. Numerical results confirm that this new algorithm is applicable. [ABSTRACT FROM AUTHOR]

Nonlinear Schrodinger equation (NLSE) is now one of the prominent of modern physics, mathematics and chemistry. Over these fields, the NLSE is also applied in new emerging fields such as quantum information and econophysics. In this paper we investigate for new exact solutions of higher order nonlinear Schrodinger's equation. This method allows to carry out the solution process of nonlinear wave equations more thoroughly and conveniently by computer algebra systems such as the Maple and Mathematica. In addition to providing a different way of solving the Schrodinger equation for such systems, the simplicity of the algorithm renders it a great pedagogical value. [ABSTRACT FROM AUTHOR]

In this paper, the first integral method is used to construct exact solutions of the time-fractional Wu-Zhang system. Fractional derivatives are described by conformable fractional derivative. This method is based on the ring theory of commutative algebra. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear conformable time-fractional partial differential equations. [ABSTRACT FROM AUTHOR]

This paper applies new conformable fractional derivative for converting fractional coupled nonlinear Schrodinger equations into the ordinary differential equations. Then by using the Kudryashov method we extract new traveling wave solutions to the fractional coupled nonlinear Schrodinger equation. This method will be analyzed for accuracy and stability. [ABSTRACT FROM AUTHOR]

This paper obtains soliton solutions in optical couplers. The governing equation is solved by the aid of G′/G-expansion scheme. There are four types of nonlinear media that are taken into consideration. These are Kerr law, power law, parabolic law, and dual-power law. There are two kinds optical couplers studied in this paper. They are twin-core couplers and multiple-core couplers, where coupling with nearest neighbors as well as coupling with all neighbors are considered. Dark and singular soliton solutions are retrieved. These soliton solutions come with constraint conditions that must hold for the solitons to exist. [ABSTRACT FROM AUTHOR]

This paper obtains bright, dark and singular soliton solutions to dense wavelength division multiplexed (DWDM) system, with spatio-temporal dispersion. There are two types of nonlinear media that are considered and they are Kerr law and parabolic law. Four integration algorithms are applied to retrieve these solitons. They are G′/G-expansion scheme, extended tanh function approach, Kudryashov's algorithm and finally the ansatz method. The results follow with respective constraints that guarantees the existence of solitons. [ABSTRACT FROM AUTHOR]

In this work, a new modified homotopy perturbation (NHPM) is applied to solve a special kind of nonlinear Volterra integral equations in two-dimensional space. The two most important steps in the application of the new Homotopy perturbation method are to construct a suitable homotopy equation and to choose a suitable initial guess. In the present paper, convergence of the new approach is proved. Comparison of our solution with the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) shows that the NHPM is effective and accurate in solving these kind of problems. [ABSTRACT FROM AUTHOR]

The Klein-Gordon equation is used to model many nonlinear phenomena. In this paper, an analytic solution of the Klein-Gordon equation is obtained by using a new modification of the homotopy perturbation method (NHPM). Theoretical considerations are discussed. The results reveal that the method is explicit, effective, and easy to use. [ABSTRACT FROM AUTHOR]

In this paper, a new form of homotopy perturbation method (NHPM) has been adopted for solving second- rder linear hyperbolic telegraph equation. This new approach leads to the exact solution of one- or two-dimensional second-order hyperbolic equation. Computations of the examples reveal the simplicity and fewer manipulations in comparison with the usual method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new homotopy perturbation method (NHPM) is introduced for obtaining solutions of systems of non-linear partial differential equations. Theoretical considerations are discussed. To illustrate the capability and reliability of the method three examples are provided. Comparison of the results of applying NHPM with those of applying HPM reveal the effectiveness and convenience of the new technique. [Copyright &y& Elsevier]

• Fast and reliable detection of HIAFs in the microgrids with renewable energy sources. • Analyzing a vast variety of useful features and their effect on the fault detection. • Using PCC and PCA, as dimension-reduction methods. • Investigating unbalanced conditions and delayed asynchrony measurements. • High performance on the networks with different configurations. High impedance Arcing faults (HIAFs) are normally caused by ruptured conductors, leaning of a tree with high impedance, and/or the presence of a high impedance object between the conductor and earth. In such cases, protections available in the microgrid may not be capable of detecting the HIAFs. Hence, to increase the safety level and reliability of the microgrid, it is essential to take action for fast and reliable detection of these types of faults. Therefore, the present study introduces an appropriate strategy to detect HIAFs using a pattern recognition approach. To this end, different scenarios are implemented in the training data extraction step considering the measurement units embedded in a 25 kV microgrid in the presence of Distributed Generations (DG) and Renewable Energy Sources (RESs) in the EMTP-RV software environment. Then, after the initial processing, the scenarios are scaled-down and compared using the Pearson Correlation Coefficient (PCC) and Principal Component Analysis (PCA) methods. Next, the processed data is classified using the Support Vector Machine (SVM) method by selecting the most appropriate kernel. Simulation results in EMTP-RV and MATLAB environments demonstrate that the proposed strategy is capable of fast detection of HIAFs in microgrids with a high level of accuracy. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is a fundamental equation in the field of nonlinear optics and plays an important role in the study of many physical phenomena. The present study introduces a new model that demonstrates the novelty of the paper and provides the advancement of knowledge in the area of nonlinear optics by solving a challenging problem known as the extended (3 + 1)-dimensional nonlinear conformable Kudryashov's equation (CKE) with generalized anti-cubic nonlinearity, which is a generalization of the NLSE to three spatial dimension and one temporal dimension for the first time. This work is significant because it advances our understanding of nonlinear optics and its applications to solve complex equations in physics and related disciplines. The extended hyperbolic function method (EHFM) and Nucci's reduction method are applied to the extended (3 + 1)-dimensional nonlinear CKE with generalized anti-cubic nonlinearity. The equation is solved by using the concept of conformable derivative, a recently developed operator in fractional calculus, which has advantages over other fractional derivatives in terms of accuracy and flexibility. The attained solutions include periodic singular, dark 1-soliton, singular 1-soliton, and bright 1-soliton which are visualized using 3D and contour plots. This study highlights the potential of using conformable derivative and the applied techniques to solve complex nonlinear differential equations in various fields. The obtained solutions and analysis will be useful in the design and analysis of optical communication systems and other related fields. Overall, this study contributes for the understanding of the dynamics of the extended (3+1)-dimensional nonlinear CKE and offers new insights into the use of mathematical techniques to tackle complex problems in physics and related fields. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to obtain the analytic solutions for stochastic fourth-order (2+1)-dimensional nonlinear Schrödinger equation (NLSE) with higher order odd and even terms. Our Study mainly depends on applying the improved modified extended tanh function method (IMETFM) to get the soliton wave solutions and other exact wave solutions of our model of interest. These solutions such as (bright, dark, singular) solitons, hyperbolic solutions, periodic solutions, singular periodic solutions, rational solutions, Jacobi elliptic functions solutions and Weierstrass elliptic functions solutions. Graphical representations of some of the extracted solutions using different noise intensities are shown to demonstrate the influence of the noise. [ABSTRACT FROM AUTHOR]

The Kerr law and dual power law Schrödinger equations for obtaining optical soliton solutions are studied. In this paper, we investigate these equations via generalized logistic equation method. These various kinds of solutions are favorable for explaining diverse nonlinear physical phenomena. The outcomes are useful in describing the diffusion of optical solitons. [ABSTRACT FROM AUTHOR]

In this paper, a numerical technique based on matrix collocation method is presented for approximating the solution of the Ambartsumian equation in terms of the shifted Bernoulli polynomials. Shifted Bernoulli operational matrix of differentiation is first derived and then utilized to reduce the problem into a system of linear algebraic equations which can be solved to find the solution. Results are included to demonstrate the validity and applicability of the technique and compared with other methods. These results show the efficiency and accuracy of the presented algorithm and also the physical behaviour of the presented problem over large intervals. [ABSTRACT FROM AUTHOR]

In this present study, new sub-equation method is used to construct exact travelling wave solutions of the Cahn-Hilliard equation (CH). As a consequence, many new general exact travelling wave solutions are obtained including generalized hyperbolic (GH) function, generalized trigonometric (GT) function, exponential function and rational function solutions. Compared with the most existing similar new sub-equation method, the proposed method gives new and more general exact travelling wave solutions. More importantly, the method with the help of symbolic computation prepares a powerful mathematical tool for solving a great many nonlinear evolution equations in mathematical physics. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of cubic–quartic NLSE with the Kerr law nonlinearity via a new sub-equation method. Considered method of undetermined coefficients is applied to obtain the introduced solutions. The outcomes are useful in describing the diffusion of optical solitons. The performance of the approach is reliable, useful and gives more new general exact solutions than the other methods. [ABSTRACT FROM AUTHOR]

In this paper, with the aid of the Maple software, the new extended direct algebraic method is used as a powerful method to constructs some new solutions to the well-known nonlinear models, namely, the simplified modified Camassa-Holm equation. The space-time fractional derivatives are defined in the sense of the new conformable fractional derivative. It is shown that the method is straightforward and effective mathematical tool for solving nonlinear evolution equations in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

In this article, an amelioration of the approaches namely the new extended direct algebraic method for solving the nonlinear conformable fractional Schrödinger-Hirota equation (FSHE) is presented. By using the traveling wave transformation change the fractional equation into the ordinary differential equation, and with the aid of mathematical software Maple, a number of new optical solitons solutions for them are calculated. The obtained results show that the proposed method is powerful, effective, and straightforward technique to work out new solutions of various types of nonlinear fractional partial differential equations in applied sciences. [ABSTRACT FROM AUTHOR]

This paper studies fractional temporal evolution of optical solitons with quadratic–cubic nonlinearity that comes with a few perturbation terms. Khalil's conformable fractional derivative as well as Liu's extended trial function scheme are applied to retrieve these soliton solutions. The results are applicable to mitigate Internet bottleneck that is a growing problem in telecommunications industry. [ABSTRACT FROM AUTHOR]

This paper studies optical solitons with fractional temporal evolution in presence of Hamiltonian perturbation terms. The three types of nonlinearity are Kerr law, parabolic law and dual-power law. The first integral method with conformable fractional derivative is applied to retrieve soliton solutions to the model. Several constraint conditions guarantee the existence of such solitons. [ABSTRACT FROM AUTHOR]

This paper utilizes trial solution algorithm to secure optical soliton solutions to the nonlinear Schrödinger’s equation. Bright, dark and singular soliton solutions are obtained to the model that is considered with four forms of nonlinearity. They are Kerr, power, parabolic and dual-power laws. There are constraint conditions that guarantee the existence of these solitons. Additionally, singular periodic solutions are revealed as a by-product of this approach, and these are also listed. [ABSTRACT FROM AUTHOR]

In this paper, new exact traveling wave solutions for the coupling BoitiLeon-Pempinelli system are obtained by using two important different methods. The first is the modified extended tanh function methods which depend on the balance rule and the second is the Ricatti-Bernoulli Sub-ODE method which doesn’t depend on the balance rule. The solitary waves solutions can be derived from the exact wave solutions by give the parameters a special value. The consistent and inconsistent of the obtained solutions are studied not only between these two methods but also with that relisted by the other methods. [ABSTRACT FROM AUTHOR]

This paper examines new travelling wave solutions to the Lakshmanan–Porsezian–Daniel (LPD) model with Kerr nonlinearity using Bäcklund transformation method based on Riccati equation, Kudryashov method and a new auxiliary ordinary differential equation (ODE). The three methods are adequately utilised, and some new rational-type hyperbolic and trigonometric function solutions are derived in different shapes for the aforementioned model. We confirm that our methods are more efficient than the other methods and it might be used in many other such types of nonlinear equations arising in the basic fabric of communications network technology and nonlinear optics. [ABSTRACT FROM AUTHOR]

The present article studies a B-type Kadomtsev-Petviashvili equation with certain applications in the fluids. Stating with the Hirota's bilinear form and adopting reliable methodologies, a group of exact solutions such as the N-wave and other solutions to the B-type Kadomtsev-Petviashvili equation is formally derived. Some figures in two and three dimensions are given to illustrate the characteristics of the obtained solutions. The results of the current work actually help to complete the previous studies about the B-type Kadomtsev-Petviashvili equation. [ABSTRACT FROM AUTHOR]

In this paper, the extended rational sinh-cosh method (ERSCM) and modified Khater method are applied to the biological population model to derive new exact solutions. Moreover, the stability property of some obtained solutions is discussed to show the ability of them for using in the model's applications. Implementation of the direct algebraic methods, the equations derived by substitution of the predicted solution are solved. It is significant to point out that new traveling wave solutions are found. The present methods are easy to employ and sufficient to determine the solutions. [ABSTRACT FROM AUTHOR]

This work presents novel perception into the optical solitons of the cubic-quartic nonlinear Schrödinger's equation (CQ-NLSE) by introducing the cubic-quintic-septic-nonic (CQSN) form of self-phase modulation (SPM). The proposed model has a significant role in physics and engineering and it is considered the equation of pulses dissemination in optical communication. To extract the optical solitons such as the bright, kink, periodic and singular solitons, the Kudryashov's method and extended hyperbolic function method (EHFM) are applied for the first time. The results show that these methods are suitable, productive and competent to find the solutions for such kind of nonlinear evolution equations (NLEEs) in optical fibers and nonlinear sciences. Moreover, for physical illustration of obtained solutions, some 3d, contour and 2d graphs are plotted which shed light on the differences between the solutions obtained through Kudryashov's method and extended hyperbolic function method (EHFM). Furthermore, comparison section is also given to prove the novelty of the work over the reported results. [ABSTRACT FROM AUTHOR]

The present study suggests a large family of optical solutions to the Kundu–Eckhaus model. Modified form of an auxiliary equation approach have been used to set the solution families. Compatible variable transform reduces the governing equation to a non linear ordinary differential equation. The parameters of the predicted solution in finite series form are determined by substitution of the solution into this ODE. The results cover a large family of optical solutions that are determined by implementation of various techniques in the literature. In this perspective, the modifed form of the a auxiliary equation approach is evaluated as a generalization of different methods widely used to construct the solutions. [ABSTRACT FROM AUTHOR]

In this paper and for the first time, we describe and introduce a new extended direct algebraic method which is a new method for solving nonlinear partial differential equations arising in nonlinear optics and nonlinear science. By applying this method, we have constructed new solitary wave solutions for the unstable Schrödinger equation. A large family of traveling wave type exact solutions covering exponential, generalized trigonometric, rational and generalized hyperbolic functions to this equation is determined. The solutions are expressed in explicit forms. [ABSTRACT FROM AUTHOR]

This study reveals the dark, bright, combined dark–bright, singular optical solitons and other solutions to the conformable space–time fractional Fokas–Lenells equation. We use two integral schemes in reaching such solutions. The constraints conditions which guarantee the existence of valid solutions are stated. By choosing the suitable values of the parameters involved, we plot the 2-dimensional and 3-dimensional surfaces to some of the obtained solutions. [ABSTRACT FROM AUTHOR]

This paper secures dark and singular resonant optical solitons that is studied with dual-power law nonlinearity and fractional temporal evolution. Khalil's conformable fractional derivative is put into perspective to retrieve these soliton solutions. The parameter restrictions for the existence of such solitons are also indicated. [ABSTRACT FROM AUTHOR]

The soliton dynamics in optical fibers with Fokas–Lenells equation is illustrated in this paper. Bright, dark and singular soliton solutions are retrieved along with few forms of combo-soliton solutions that also naturally emerged from the three integration schemes applied to the model. The existence criteria of these solitons are also presented. [ABSTRACT FROM AUTHOR]

The generalized projective Riccati equation method is proposed to establish exact solutions for generalized form of the reaction Duffing model in fractional sense namely, Khalil’s derivative. The compatible traveling wave transform converts the governing equation to a non linear ODE. The predicted solution is a series of two new variables that solve a particular ODE system. Coefficients of terms in the series are calculated by solving an algebraic system that comes into existence by substitution of the predicted solution into the ODE which is the result of the wave transformation of the governing equation. Returning original variables give exact solutions to the governing equation in various forms. [ABSTRACT FROM AUTHOR]