On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation*Mathematics* **2017**, *5*(4), 76; doi:10.3390/math5040076 - 8 December 2017**Abstract **

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral

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In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.
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Acting Semicircular Elements Induced by Orthogonal Projections on Von-Neumann-Algebras*Mathematics* **2017**, *5*(4), 74; doi:10.3390/math5040074 - 6 December 2017**Abstract **

In this paper, we construct a free semicircular family induced by $\left|\mathbb{Z}\right|$ -many mutually-orthogonal projections, and construct Banach *-probability spaces containing the family, called the free filterizations. By acting a free filterization on fixed von Neumann algebras, we construct the corresponding Banach *-probability

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In this paper, we construct a free semicircular family induced by $\left|\mathbb{Z}\right|$ -many mutually-orthogonal projections, and construct Banach *-probability spaces containing the family, called the free filterizations. By acting a free filterization on fixed von Neumann algebras, we construct the corresponding Banach *-probability spaces, called affiliated free filterizations. We study free-probabilistic properties on such new structures, determined by both semicircularity and free-distributional data on von Neumann algebras. In particular, we study how the freeness on free filterizations, and embedded freeness conditions on fixed von Neumann algebras affect free-distributional data on affiliated free filterizations.
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Geometric Structure of the Classical Lagrange-d’Alambert Principle and its Application to Integrable Nonlinear Dynamical Systems*Mathematics* **2017**, *5*(4), 75; doi:10.3390/math5040075 - 5 December 2017**Abstract **

The classical Lagrange-d’Alembert principle had a decisive influence on formation of modern analytical mechanics which culminated in modern Hamilton and Poisson mechanics. Being mainly interested in the geometric interpretation of this principle, we devoted our review to its deep relationships to modern Lie-algebraic

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The classical Lagrange-d’Alembert principle had a decisive influence on formation of modern analytical mechanics which culminated in modern Hamilton and Poisson mechanics. Being mainly interested in the geometric interpretation of this principle, we devoted our review to its deep relationships to modern Lie-algebraic aspects of the integrability theory of nonlinear heavenly type dynamical systems and its so called Lax-Sato counterpart. We have also analyzed old and recent investigations of the classical M. A. Buhl problem of describing compatible linear vector field equations, its general M.G. Pfeiffer and modern Lax-Sato type special solutions. Especially we analyzed the related Lie-algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Plebański and later analyzed in a series of articles. As effective tools the AKS-algebraic and related $\mathcal{R}$ -structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is also shown that all these equations originate in this way and can be represented as a Lax-Sato compatibility condition for specially constructed loop vector fields on the torus. Typical examples of such heavenly type equations, demonstrating in detail their integrability via the scheme devised herein, are presented.
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Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation*Mathematics* **2017**, *5*(4), 73; doi:10.3390/math5040073 - 4 December 2017**Abstract **

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The high gain free electron laser (FEL) equation is a Volterra type integro-differential equation amenable for analytical solutions in a limited number of cases. In this note, a novel technique, based on an expansion employing a family of two variable Hermite polynomials, is

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The high gain free electron laser (FEL) equation is a Volterra type integro-differential equation amenable for analytical solutions in a limited number of cases. In this note, a novel technique, based on an expansion employing a family of two variable Hermite polynomials, is shown to provide straightforward analytical solutions for cases hardly solvable with conventional means. The possibility of extending the method by the use of expansion using different polynomials (two variable Legendre like) expansion is also discussed.
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Some Types of Subsemigroups Characterized in Terms of Inequalities of Generalized Bipolar Fuzzy Subsemigroups*Mathematics* **2017**, *5*(4), 71; doi:10.3390/math5040071 - 27 November 2017**Abstract **

In this paper, we introduce a generalization of a bipolar fuzzy (BF) subsemigroup, namely, a $({\alpha}_{1},{\alpha}_{2};{\beta}_{1},{\beta}_{2})$ -BF subsemigroup. The notions of $({\alpha}_{1},{\alpha}_{2};{\beta}_{1},{}_{}$

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In this paper, we introduce a generalization of a bipolar fuzzy (BF) subsemigroup, namely, a $({\alpha}_{1},{\alpha}_{2};{\beta}_{1},{\beta}_{2})$ -BF subsemigroup. The notions of $({\alpha}_{1},{\alpha}_{2};{\beta}_{1},{\beta}_{2})$ -BF quasi(generalized bi-, bi-) ideals are discussed. Some inequalities of $({\alpha}_{1},{\alpha}_{2};{\beta}_{1},{\beta}_{2})$ -BF quasi(generalized bi-, bi-) ideals are obtained. Furthermore, any regular semigroup is characterized in terms of generalized BF semigroups.
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Wavelet Neural Network Model for Yield Spread Forecasting*Mathematics* **2017**, *5*(4), 72; doi:10.3390/math5040072 - 27 November 2017**Abstract **

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In this study, a hybrid method based on coupling discrete wavelet transforms (DWTs) and artificial neural network (ANN) for yield spread forecasting is proposed. The discrete wavelet transform (DWT) using five different wavelet families is applied to decompose the five different yield spreads

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In this study, a hybrid method based on coupling discrete wavelet transforms (DWTs) and artificial neural network (ANN) for yield spread forecasting is proposed. The discrete wavelet transform (DWT) using five different wavelet families is applied to decompose the five different yield spreads constructed at shorter end, longer end, and policy relevant area of the yield curve to eliminate noise from them. The wavelet coefficients are then used as inputs into Levenberg-Marquardt (LM) ANN models to forecast the predictive power of each of these spreads for output growth. We find that the yield spreads constructed at the shorter end and policy relevant areas of the yield curve have a better predictive power to forecast the output growth, whereas the yield spreads, which are constructed at the longer end of the yield curve do not seem to have predictive information for output growth. These results provide the robustness to the earlier results.
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Controlling Chaos—Forced van der Pol Equation*Mathematics* **2017**, *5*(4), 70; doi:10.3390/math5040070 - 24 November 2017**Abstract **

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Nonlinear systems are typically linearized to permit linear feedback control design, but, in some systems, the nonlinearities are so strong that their performance is called chaotic, and linear control designs can be rendered ineffective. One famous example is the van der Pol equation

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Nonlinear systems are typically linearized to permit linear feedback control design, but, in some systems, the nonlinearities are so strong that their performance is called chaotic, and linear control designs can be rendered ineffective. One famous example is the van der Pol equation of oscillatory circuits. This study investigates the control design for the forced van der Pol equation using simulations of various control designs for iterated initial conditions. The results of the study highlight that even optimal linear, time-invariant (LTI) control is unable to control the nonlinear van der Pol equation, but idealized nonlinear feedforward control performs quite well after an initial transient effect of the initial conditions. Perhaps the greatest strength of ideal nonlinear control is shown to be the simplicity of analysis. Merely equate coefficients order-of-differentiation insures trajectory tracking in steady-state (following dissipation of transient effects of initial conditions), meanwhile the solution of the time-invariant linear-quadratic optimal control problem with infinite time horizon is needed to reveal constant control gains for a linear-quadratic regulator. Since analytical development is so easy for ideal nonlinear control, this article focuses on numerical demonstrations of trajectory tracking error.
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Impact of Parameter Variability and Environmental Noise on the Klausmeier Model of Vegetation Pattern Formation*Mathematics* **2017**, *5*(4), 69; doi:10.3390/math5040069 - 23 November 2017**Abstract **

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Semi-arid ecosystems made up of patterned vegetation, for instance, are thought to be highly sensitive. This highlights the importance of understanding the dynamics of the formation of vegetation patterns. The most renowned mathematical model describing such pattern formation consists of two partial differential

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Semi-arid ecosystems made up of patterned vegetation, for instance, are thought to be highly sensitive. This highlights the importance of understanding the dynamics of the formation of vegetation patterns. The most renowned mathematical model describing such pattern formation consists of two partial differential equations and is often referred to as the Klausmeier model. This paper provides analytical and numerical investigations regarding the influence of different parameters, including the so-far not contemplated evaporation, on the long-term model results. Another focus is set on the influence of different initial conditions and on environmental noise, which has been added to the model. It is shown that patterning is beneficial for semi-arid ecosystems, that is, vegetation is present for a broader parameter range. Both parameter variability and environmental noise have only minor impacts on the model results. Increasing mortality has a high, nonlinear impact underlining the importance of further studies in order to gain a sufficient understanding allowing for suitable management strategies of this natural phenomenon.
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Channel Engineering for Nanotransistors in a Semiempirical Quantum Transport Model*Mathematics* **2017**, *5*(4), 68; doi:10.3390/math5040068 - 22 November 2017**Abstract **

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One major concern of channel engineering in nanotransistors is the coupling of the conduction channel to the source/drain contacts. In a number of previous publications, we have developed a semiempirical quantum model in quantitative agreement with three series of experimental transistors. On the

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One major concern of channel engineering in nanotransistors is the coupling of the conduction channel to the source/drain contacts. In a number of previous publications, we have developed a semiempirical quantum model in quantitative agreement with three series of experimental transistors. On the basis of this model, an overlap parameter $0\le C\le 1$ can be defined as a criterion for the quality of the contact-to-channel coupling: A high level of *C* means good matching between the wave functions in the source/drain and in the conduction channel associated with a low contact-to-channel reflection. We show that a high level of *C* leads to a high saturation current in the ON-state and a large slope of the transfer characteristic in the OFF-state. Furthermore, relevant for future device miniaturization, we analyze the contribution of the tunneling current to the total drain current. It is seen for a device with a gate length of 26 nm that for all gate voltages, the share of the tunneling current becomes small for small drain voltages. With increasing drain voltage, the contribution of the tunneling current grows considerably showing Fowler–Nordheim oscillations. In the ON-state, the classically allowed current remains dominant for large drain voltages. In the OFF-state, the tunneling current becomes dominant.
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Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results*Mathematics* **2017**, *5*(4), 65; doi:10.3390/math5040065 - 21 November 2017**Abstract **

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With fractional differential equations (FDEs) rising in popularity and methods for solving them still being developed, approximations to solutions of fractional initial value problems (IVPs) have great applications in related fields. This paper proves an extension of Picard’s Iterative Existence and Uniqueness Theorem

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With fractional differential equations (FDEs) rising in popularity and methods for solving them still being developed, approximations to solutions of fractional initial value problems (IVPs) have great applications in related fields. This paper proves an extension of Picard’s Iterative Existence and Uniqueness Theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous term satisfies the usual Lipschitz’s condition. As an application of our method, we have provided several numerical examples.
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On Edge Irregular Reflexive Labellings for the Generalized Friendship Graphs*Mathematics* **2017**, *5*(4), 67; doi:10.3390/math5040067 - 21 November 2017**Abstract **

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We study an edge irregular reflexive *k*-labelling for the generalized friendship graphs, also known as flowers (a symmetric collection of cycles meeting at a common vertex), and determine the exact value of the reflexive edge strength for several subfamilies of the generalized

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We study an edge irregular reflexive *k*-labelling for the generalized friendship graphs, also known as flowers (a symmetric collection of cycles meeting at a common vertex), and determine the exact value of the reflexive edge strength for several subfamilies of the generalized friendship graphs.
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Generalized Langevin Equation and the Prabhakar Derivative*Mathematics* **2017**, *5*(4), 66; doi:10.3390/math5040066 - 20 November 2017**Abstract **

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We consider a generalized Langevin equation with regularized Prabhakar derivative operator. We analyze the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. We further introduce the so-called *tempered* regularized Prabhakar derivative and analyze the corresponding generalized Langevin equation with friction term

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We consider a generalized Langevin equation with regularized Prabhakar derivative operator. We analyze the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. We further introduce the so-called *tempered* regularized Prabhakar derivative and analyze the corresponding generalized Langevin equation with friction term represented through the tempered derivative. Various diffusive behaviors are observed. We show the importance of the three parameter Mittag-Leffler function in the description of anomalous diffusion in complex media. We also give analytical results related to the generalized Langevin equation for a harmonic oscillator with generalized friction. The normalized displacement correlation function shows different behaviors, such as monotonic and non-monotonic decay without zero-crossings, oscillation-like behavior without zero-crossings, critical behavior, and oscillation-like behavior with zero-crossings. These various behaviors appear due to the friction of the complex environment represented by the Mittag-Leffler and tempered Mittag-Leffler memory kernels. Depending on the values of the friction parameters in the system, either diffusion or oscillations dominate.
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On the Inception of Financial Representative Bubbles*Mathematics* **2017**, *5*(4), 64; doi:10.3390/math5040064 - 17 November 2017**Abstract **

In this work, we aim to formalize the inception of representative bubbles giving the condition under which they may arise. We will find that representative bubbles may start at any time, depending on the definition of a behavioral component. This result is at

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In this work, we aim to formalize the inception of representative bubbles giving the condition under which they may arise. We will find that representative bubbles may start at any time, depending on the definition of a behavioral component. This result is at odds with the theory of classic rational bubbles, which are those models that rely on the fulfillment of the transversality condition by which a bubble in a financial asset can arise just at its first trade. This means that a classic rational bubble (differently from our model) cannot follow a cycle since if a bubble exists, it will burst by definition and never arise again.
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Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations*Mathematics* **2017**, *5*(4), 63; doi:10.3390/math5040063 - 16 November 2017**Abstract **

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Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can

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Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can be designed for an arbitrary order of accuracy. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. We analyze the truncation errors of the fully discretized schemes. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs.
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Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function*Mathematics* **2017**, *5*(4), 62; doi:10.3390/math5040062 - 10 November 2017**Abstract **

The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green’s function are obtained in the framework of distribution theory. In particular, discussions are given on Kummer’s and the hypergeometric differential equation. Related discussions are given on the particular

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The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green’s function are obtained in the framework of distribution theory. In particular, discussions are given on Kummer’s and the hypergeometric differential equation. Related discussions are given on the particular solution of differential equations with constant coefficients, by the Laplace transform.
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Mixed Order Fractional Differential Equations*Mathematics* **2017**, *5*(4), 61; doi:10.3390/math5040061 - 7 November 2017**Abstract **

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.
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Graph Structures in Bipolar Neutrosophic Environment*Mathematics* **2017**, *5*(4), 60; doi:10.3390/math5040060 - 6 November 2017**Abstract **

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A bipolar single-valued neutrosophic (BSVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of BSVN graph structures. We describe some operations on BSVN graph structures and elaborate on these with examples. Moreover, we

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A bipolar single-valued neutrosophic (BSVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of BSVN graph structures. We describe some operations on BSVN graph structures and elaborate on these with examples. Moreover, we investigate some related properties of these operations.
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A Constructive Method for Standard Borel Fixed Submodules with Given Extremal Betti Numbers*Mathematics* **2017**, *5*(4), 56; doi:10.3390/math5040056 - 1 November 2017**Abstract **

Let *S* be a polynomial ring in *n* variables over a field *K* of any characteristic. Given a strongly stable submodule *M* of a finitely generated graded free *S*-module *F*, we propose a method for constructing a standard Borel-fixed submodule $\stackrel{}{M}$

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Let *S* be a polynomial ring in *n* variables over a field *K* of any characteristic. Given a strongly stable submodule *M* of a finitely generated graded free *S*-module *F*, we propose a method for constructing a standard Borel-fixed submodule $\tilde{M}$ of *F* so that the extremal Betti numbers of *M*, values as well as positions, are preserved by passing from *M* to $\tilde{M}$ . As a result, we obtain a numerical characterization of all possible extremal Betti numbers of any standard Borel-fixed submodule of a finitely generated graded free *S*-module *F*.
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The Theory of Connections: Connecting Points*Mathematics* **2017**, *5*(4), 57; doi:10.3390/math5040057 - 1 November 2017**Abstract **

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This study introduces a procedure to obtain *all* interpolating functions, $y=f\left(x\right)$ , subject to linear constraints on the function and its derivatives defined at specified values. The paper first shows how to express these interpolating functions passing through

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This study introduces a procedure to obtain *all* interpolating functions, $y=f\left(x\right)$ , subject to linear constraints on the function and its derivatives defined at specified values. The paper first shows how to express these interpolating functions passing through a single point in three distinct ways: linear, additive, and rational. Then, using the additive formalism, interpolating functions with linear constraints on one, two, and *n* points are introduced as well as those satisfying relative constraints. In particular, for expressions passing through *n* points, a generalization of the Waring’s interpolation form is introduced. An alternative approach to derive additive *constraint interpolating expressions* is introduced requiring the inversion of a matrix with dimensions equally the number of constraints. Finally, continuous and discontinuous interpolating periodic functions passing through a set of points with specified periods are provided. This theory has already been applied to obtain least-squares solutions of initial and boundary value problems applied to nonhomogeneous linear differential equations with nonconstant coefficients.
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Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations*Mathematics* **2017**, *5*(4), 59; doi:10.3390/math5040059 - 1 November 2017**Abstract **

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Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method.
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