Search

This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method. In this method, utilizing a traveling wave transformation with the aid of the Riccati-Bernoulli equation, the fourth-order equation can be transformed into a set of algebraic equations. Solving the set of algebraic equations, we acquire the novel exact solutions of the integrable fourth-order equations presented in this research paper. The physical interpretation of the nonlinear models are also detailed through the exact solutions, which demonstrate the effectiveness of the presented method.The Bäcklund transformation can produce an infinite sequence of solutions of the given two fourth-order nonlinear partial differential equations. Finally, 3D graphs of some derived solutions in this paper are depicted through suitable parameter values.

International audience; We explore the spatial spread of vector-borne infections with conditional vector preferences, meaning that vectors do not visit hosts at random. Vectors may be differentially attracted toward infected and uninfected hosts depending on whether they carry the pathogen or not. The model is expressed as a system of partial differential equations with vector diffusion. We first study the diffusion-less model. We show that conditional vector preferences alone (in the absence of any epidemiological feedback on their population dynamics) may result in bistability between the disease-free equilibrium and an endemic equilibrium. A backward bifurcation may allow the disease to persist even though its basic reproductive number is less than one. Bistability can occur only if both infected and uninfected vectors prefer uninfected hosts. Back to the model with diffusion, we show that bistability in the local dynamics may generate travelling waves with either positive or negative spreading speeds, meaning that the disease either invades or retreats into space. In the monostable case, we show that the disease spreading speed depends on the preference of uninfected vectors for infected hosts but not on the preference of infected vectors for uninfected hosts. We discuss the implications of our results for vector-borne plant diseases, which are the main source of evidence for conditional vector preferences so far.

In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

This work investigates the complex Ginzburg–Landau equation (CGLE) with parabolic law in nonlinear optics, this form of nonlinearity may also be seen in fiber optics. It is referred to as the fifth-order susceptibility, which is predominantly present in a transparent glass with intense femtosecond pulses at 620nmI. The ϕ6-model expansion approach is used to find dark, singular, periodic, and combined optical soliton solutions to the model. The results presented in this study are intended to improve the CGLE’s nonlinear dynamical characteristics. These solitons are significant resources in physics and telecommunications engineering. They led to several quick follow-up investigations. The hyperbolic sine, for example, appears in the calculation of the Roche limit and gravitational potential of a cylinder, while the hyperbolic cotangent appears in the Langevin function for magnetic polarization. Some of the obtained solutions’ 2-, 3-dimensional, and contour plots are shown.

In this paper, within the scope of the local fractional derivative theory, bidirectional propagation system local fractional equations are researched. Compared with the unidirectional propagation of nonlinear waves in a pipeline, the bidirectional propagation system equations studied in this paper can better describe the propagation of nonlinear waves in a channel. This study is groundbreaking and offers more possibilities for the bidirectional propagation of nonlinear waves in the simulation pipeline. The exact traveling wave solutions of the non-differentiable type defined on the Cantor sets are obtained. The characteristics of the particular solutions of a fixed fractal dimension are discussed. It is proven that the local fractional nonlinear bidirectional wave equations can describe the interaction of fractal waves. It is also shown that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics.

In this study, some new exact wave solutions of nonlinear partial differential equations are investigated by the modified simple equation method. This method is applied to the $(2+1)$-dimensional Calogero-Bogoyavlenskii-Schiff equation and the $(3+1)$-dimensional Jimbo-Miwa equation. Our applications reveal how to use the proposed method to solve nonlinear partial differential equations with the balance number equal to two. Consequently, some new exact traveling wave solutions of these equations are achieved, and types of waves are determined. To verify our results and draw the graphs of the solutions, we use the Mathematica package program.

This paper considers the generalized Davey-Stewartson equation that is used to investigate the dynamics of wave propagation in water of finite depth under the effects of gravity force and surface tension. The model is considered in the presence of full nonlinearity. The main objective of this paper is to extract soliton solutions of the generalized Davey-Stewartson equation. Three state-of-the-art integration schemes, namely exp ( - Φ ( ξ ) ) -expansion method, the first integral method and the Sine-Gordon expansion method have been employed for obtaining the desired soliton solutions. The proposed methods successfully attain different structures of explicit solutions such as bright, dark, singular, rational and periodic solitary wave solutions. All the newly found solutions are discussed with their existence criteria. The 2D and 3D portraits are also shown for some of the reported solutions.

The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi’s elliptic function expansion. The presented coupled KdV equations have a more general form than those studied in the literature. Nine couples of quadratic Jacobi’s elliptic function solutions are found. Each couple of traveling wave solutions is symmetric in mathematical form. In the limit cases m→1, these periodic solutions degenerate as the corresponding soliton solutions. After the simple parameter substitution, the trigonometric function solutions are also obtained.

The main goal of this study is obtaining analytical solutions for (3+1)-dimensional Jimbo-Miwa Equation which the second equation in the well-known KP hierarchy of integrable systems. For the (3+1DJM) equation, hyperbolic, trigonometric, complex trigonometric and rational traveling wave solutions have been constructed by applying the (G'/G,1/G)-expansion method. Then, real and imaginary graphics are presented by giving special values to the constants in the solutions obtained. These graphics are a special solution of the (3+1DJM) equation and represent a stationary wave of the equation. The (G'/G,1/G)-expansion method is an effective and powerful method for solving nonlinear evolution equations (NLEEs). Ready computer package program is used to obtain the solutions and graphics presented in this study.

In this paper, the coupled local fractional sine-Gordon equations are studied in the range of local fractional derivative theory. The study of exact solutions of nonlinear coupled systems is of great significance for understanding complex physical phenomena in reality. The main method used in this paper is the local fractional homotopy perturbation method, which is used to analyze the exact traveling wave solutions of generalized nonlinear systems defined on the Cantor set in the fractal domain. The fractal wave with fractal dimension ε=ln2/ln3 is numerically simulated. Through numerical simulation, we find that the obtained solutions are of great significance to explain some practical physical problems.

In this manuscript, exact solutions of the Oskolkov equation, which describes the dynamics of incompressible viscoelastic Kelvin-Voigt fluid, are presented. The -expansion method is used to search for these solutions. The dynamics of the obtained exact solutions are analyzed with the help of appropriate parameters and presented with graphics. The applied method is efficient and reliable to search for fundamental nonlinear waves that enrich the various dynamical models seen in engineering fields. It is concluded that the analytical method used in the study of the Oskolkov equation is reliable, valid and useful tool for created traveling wave solutions., Bu çalışmada, sıkıştırılamaz visko-elastik Kelvin-Voigt akışkanının dinamiklerini tanımlayan Oskolkov denkleminin tam çözümleri sunulmuştur. Bu çözümleri aramak için -açılım yöntemi kullanılmaktadır. Elde edilen tam çözümlerinin dinamikleri uygun parametreler yardımıyla analiz edilmiş ve grafiklerle sunulmuştur. Uygulanan yöntem, mühendislik alanlarında görülen çeşitli dinamik modelleri zenginleştiren temel doğrusal olmayan dalgaları aramak için etkili ve güvenilirdir. Oskolkov denkleminin çalşmasında kullanılan analitik metodun gezici dalga çözümlerini ortaya koymakta güvenilir, geçerli ve faydalı bir araç olduğu sonucu elde edilir.

Various collective motions of camphor boats have been studied. Camphor boats are self-driven particles that interact with each other through the change of surface tension on water surface by camphor molecules. Consequently, even in a one-dimensional circuit, a congested state or jamming can be observed (Suematsu et al. in Phys Rev E 81:056210, 2010). In this phenomenon, the unidirectional motion of each particle is considered a traveling wave, and the concentration of camphor molecules forms a pulse shape. Hence, each pair of particles interacts with each other like a pulse–pulse interaction. Thus, we expect that the center manifold theories proposed in Ei (J Dyn Differ Equ 14:85–137, 2002) and Ei et al. (Physica D 165:176–198 2002) are applicable for the analysis of the collective motion of camphor boats. However, spatial discontinuity in our model, in particular the existence of Dirac delta functions in a linearized operator, does not fulfill the requirement in the reduction process because the authors developed their theories in $$L^2$$ -framework and for smooth nonlinearity in Ei (2002) and Ei et al. (2002). In this article, we extend the results obtained in Ei (2002) and Ei et al. (2002) and establish a new center manifold approach in $$(H^1)^*$$ -framework. Finally, we succeed to rigorously reduce a mathematical model (Nagayama et al. in Physica D: Nonlinear Phenom 194:151–165, 2004) coupled with a Newtonian equation and a reaction–diffusion equation including delta functions to an ordinary differential system that represents the motions of camphor boats.

In this paper, sub equation and ð Þ 1=G’ expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation (LWE) in a magneto-electro-elastic circular rod. The proposed methods have been used to construct hyperbolic, rational, dark soliton and trigonometric solutions of the LWE in the magnetoelectro-elastic circular rod. Arbitrary values are given to the parameters in the solutions obtained. 3D, 2D and contour graphs are presented with the help of a computer package program. Solutions attained by symbolic calculations revealed that these methods are effective, reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics.

In this paper, we investigate the conformable time fractional Burgers type equations. First, we construct the explicit solutions of Riccati equation by means of modified tanh function method and modified extended exp-function method respectively. In addition, based on the formulas obtained above, the traveling wave solutions of conformable time fractional Burgers equation and (2+1)-dimensional generalized conformable time fractional Burgers equations are established applying functional separation variables method. Furthermore, the three-dimensional diagrams of the obtained exact solutions are presented for the purpose of visualization.

In this paper, a family of modified Korteweg-de Vries equations within local fractional derivative are constructed, and their non-differentiable solutions are dis-cussed by using several methods.

Mathematical modeling for cancerous disease has attracted increasing attention from the researchers around the world. Being an effective tool, it helps to describe the processes that happen to the tumour as the diverse treatment scenarios. In this paper, a density-dependent reaction-diffusion equation is applied to the most aggressive type of brain cancer, Glioblastoma multiforme. The model contains the terms responsible for the growth, migration and proliferation of the malignant tumour. The traveling wave solution used is justified by stability analysis. Numerical simulation of the model is provided and the results are compared with the experimental data obtained from the reference papers.

This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source, $$\displaylines{ u_t=\Delta u -\chi\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N,\cr \tau v_t=\Delta v-\lambda v +\mu u,\quad x\in\mathbb{R}^N, }$$ where $\chi, \mu,\lambda,a,b$ are positive numbers, and $\tau\ge 0$. Among others, it is proved that if $b>2\chi\mu$ and $\tau \geq \frac{1}{2}(1-\frac{\lambda}{a})_{+}$, then for every $c\ge 2\sqrt{a}$, this system has a traveling wave solution $(u,v)(t,x)=(U^{\tau,c}(x\cdot\xi-ct),V^{\tau,c}(x\cdot\xi-ct))$ (for all $\xi\in\mathbb{R}^N $) connecting the two constant steady states $(0,0)$ and $(\frac{a}{b},\frac{\mu}{\lambda}\frac{a}{b})$, and there is no such solutions with speed $c$ less than $2\sqrt{a}$, which improves the results established in [30] and shows that this system has a minimal wave speed $c_0^*=2\sqrt a$, which is independent of the chemotaxis.

In this paper, we apply the two variable (φ'/φ, 1/φ)-expansion method to seek exact traveling wave solutions (solitary wave solutions, periodic function solutions, rational function solution) for time-fractional Kuramoto-Sivashinsky (K-S) equation, (3+1)-dimensional time-fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation and time-fractional Sharma-Tasso-Olver (FSTO) equation. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions containing parameters. The results show that the two variable (φ'/φ, 1/φ)-expansion method is simple, effctivet, straightforward and is the generalization of the (G'/G)-expansion method.

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

We opted to construct a traveling wave solution to the nonlinear space-time fractional coupled modified equal width (CMEW) equation and the space-time fractional-coupled Burgers equation, which are often used as an electro-hydro-dynamical model to advance the local electric field and particle acoustic waves in plasma, the shallow water wave issues and portray the variety additional time of an actual structure on the partial liquid mechanics framework, particle acoustic waves through a gas-filled line, and certain consistent state gooey liquid. In this study, we employ the two variable (G /G, 1/G)-expansion method to create further general solitary wave solutions to those equations based on Riemann– Liouville fractional derivative. The fractional differential wave transform simplifies by generating ordinary differential equations (ODE) from fractional-order differential equations. We identified multiple types of solutions through the maple that are illustrated using 3D shape, 3D list point plot, and contour narratives. Additionally, we proposed that the methodology be changed to be more pragmatic, economical, and dependable and that we investigate more generalized precise solutions for traveling waves, such as solitary wave solutions.

International audience; We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions.

In this article, we investigate the lump, soliton, periodic, kink, and rogue waves to the time-fractional phi-four and (2+1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equations. The (G′/G,1/G)-expansion technique is applied to obtain the solutions of the phi-four and Calogero-Bogoyavlanskil schilf (CBS) equations with the contribution of conformable derivative. The stated method declines a huge volume of calculation. The (G′/G,1/G)-expansion technique is also known as the two-variable method. Consequently, lump, mixed lump, soliton, bell-shaped soliton, periodic, kink, periodic-kink, singular kink, rogue wave solutions are displayed in hyperbolic and trigonometric function solutions. Varieties of other traveling wave solutions are also formulated with the mentioned method. Moreover, for the physical explanation of the obtained solutions, the three-dimensional (3D) surface plots, contour plots combined with 3D surface plots, and two-dimensional (2D) plots are presented using the computer software MATLAB. Additionally, it is possible to deduce that the obtained solutions and their physical properties will aid in the understanding of water wave circulation in nonlinear dynamics.

We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.

In the present paper, we firstly improve the results on traveling wave solution that were established in (Liu and Weng in J. Differ. Equ. 258:3688–3741, 2015) for a neutral reaction–diffusion equation with quasi-monotone reaction. Secondly, by constructing two auxiliary equations and using Schauder’s Fixed Point Theorem, we further establish the existence and the asymptotic properties of the traveling wave solution for the equation with non-monotone reaction. Two examples are also given as the application of our results.

A monotone iteration scheme for traveling waves based on ordered upper and lower solutions is derived for a class of nonlocal dispersal system with delay. Such system can be used to study the competition among nonlocally diffusive species and degenerately diffusive species. An example of such system is studied in detail. We show the existence of the traveling wave solutions for this system by this iteration scheme. In addition, we study the minimal wave speed, uniqueness, strict monotonicity and asymptotic behavior of the traveling wave solutions.

This article concerns a three-component delayed lattice dynamical system arising in competition models. In such models, traveling wave solutions serve an important tool to understand the competition mechanism, i.e. which species will survive or die out eventually. We first prove the existence of the minimal wave speed of the traveling wavefronts connecting two equilibria (1,0,1) and (0,1,0). Then, for sufficiently small intra-specific competitive delays, we establish the asymptotic behavior of the traveling wave solutions at minus/plus infinity. Finally the strict monotonicity and uniqueness of all traveling wave solutions are obtained for the case where intra-specific competitive delays are zeros. In particular, the effect of the delays on the minimal wave speed and the decay rate of the traveling profiles at minus/plus infinity is also investigated.

This paper considers methods to extract exact, explicit, and new single soliton solutions related to the nonlinear Klein-Gordon-Schrödinger model that is utilized in the study of neutral scalar mesons associated with conserved scalar nucleons coupled through the Yukawa interaction. Three state of the art integration schemes, namely, the e−Φ(ξ)-expansion method, Kudryashov's method, and the tanh-coth expansion method are employed to extract bright soliton, dark soliton, periodic soliton, combo soliton, kink soliton, and singular soliton solutions. All the constructed solutions satisfy their existence criteria. It is shown that these methods are concise, straightforward, promising, and reliable mathematical tools to untangle the physical features of mathematical physics equations.

In this paper we investigate the existence of traveling wave for a one-dimensional reaction diffusion system. We show that this system has a unique translation traveling wave solution.

This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $ c^* $ such that the system possesses a traveling wave solution for any given $ c > c^* $. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $ c = c^* $ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $ c < c^* $ is also discussed.

Analytic solutions of fractional order physical equations are very significant to explain the behavior of mathematical models expressing complex phenomena in engineering and natural sciences. The modified extended tanh-function (METHF) method is an especially capable and highly effective mathematical technique to attain analytic traveling wave solutions. This research proposes to examine the analytic solutions of the time-fractional (2 + 1)-dimensional non-linear Heisenberg ferromagnetic spin chain (HFSC) equation that describes electromagnetic waves in modern magnet theory by using the suggested method and the definition of conformable fractional derivative. We obtain some new analytic solutions of the proposed equation in terms of hyperbolic, trigonometric, and rational functions. The validity and precision of these solutions are also examined. The 2D, 3D, and contour graphs of solutions are given to manifest the physical behavior of the waves with the aid of the Mathematica package program. (C) 2021 Elsevier B.V. All rights reserved.

We analyze the paper “Construction of traveling waves patterns of (1+n)-dimensional modified Zakharov-Kuznetsov equation in plasma physics” by Adil Jhangeer and et al. [RINP, 2020, 103330] and demonstrate that results of this paper are wrong. The correct results are given.

This paper is concerned with the stability of an age-structured susceptible&ndash, exposed&ndash, infective&ndash, recovered&ndash, susceptible (SEIRS) model with time delay. Firstly, the traveling wave solution of system can be obtained by using the method of characteristic. The existence and uniqueness of the continuous traveling wave solution is investigated under some hypotheses. Moreover, the age-structured SEIRS system is reduced to the nonlinear autonomous system of delay ODE using some insignificant simplifications. It is studied that the dimensionless indexes for the existence of one disease-free equilibrium point and one endemic equilibrium point of the model. Furthermore, the local stability for the disease-free equilibrium point and the endemic equilibrium point of the infection-induced disease model is established. Finally, some numerical simulations were carried out to illustrate our theoretical results.

We study bifurcation of traveling wave solutions of a class of (3+1)-dimensional nonlinear evolution equations generated by the Jaulent-Miodek hierarchy. We obtain phase portraits of the nonlinear transformation system according to the different bifurcation regions of parameters. Different kinds of traveling wave solutions, such as the periodic wave solutions, solitary wave solutions, kink wave solutions and anti-kink wave solutions are found to exist under certain parameter conditions, and the exact solutions of traveling waves are obtained.

By the Wiman–Valiron theory and the complex method, we prove that all meromorphic solutions of the fifth-order ODE u n u ′ ′ ′ ′ ′ − c u = 0 belong to W , and obtain several new solutions of the sixth-order thin-film equation comparing to the opening literature. At last we propose two unsolved problems for the readers.

We study the existence of traveling wave solutions in a higher-dimensional lattice cooperative system with stage structure. We establish the existence theorem of traveling wave solutions based on the upper and lower solutions method and Schauder's fixed point theorem. Then we construct a pair of upper and lower solutions to verify the existence of traveling wave solutions.

So far, mobility in fish population has not been given sufficient attention, although movement and spatial heterogeneity can be an important dynamic feature that plays a critical role in their exploitation. Having a qualitative framework to describe and estimate movement and growth based on tagging data is necessary for efficient control and management of the fishing industry. Here, we construct an advection-diffusion-reaction model for the fish population structured to track the population densities of both the tagged fish and the tag-free fish, in which the impulsive tagging practice is incorporated, while continuous tagging is assumed to be done on off springs of trackable tagged fish, or on those in the same swarm as the trackable tagged fish. Using the traveling wave coordinate, we derive analytical expression for the solutions to the model system. We derive the explicit expression for the level of tagged fish which increases in a periodic impulsive fashion. Stability and phase plane analyses are also carried out to determine different behavior permitted by the model system.

In the paper, we study a plasma fluid physical model, namely the Zakharov–Kuznetsov (ZK, for simplicity) equation with fractional power nonlinear terms by the complete discrimination system for polynomial method, and give a detailed construction of all its single traveling wave solutions. The results show abundant traveling wave patterns of the ZK equation.

The space time fractional modified regularized long wave equation is a model equation to the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, the hydro-magnetic waves in cold plasma, the phonetic waves in dissident quartz and phonetic gravitational waves in contractible liquids. In nonlinear science and engineering, the mentioned equation is applied to analyze the one way tract of long waves in seas and harbors. In this study, the closed form traveling wave solutions to the above equation are evaluated due to conformable fractional derivatives through double (G'⁄G,1⁄G)-expansion method and the Exp-function method. The existence of chain rule and the derivative of composite function permit the nonlinear fractional differential equations (NLFDEs) converted into ODEs using wave transformation. The obtain solutions are very much effective to analyze the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, one way tract of long waves in seas and harbors. These two methods are efficient, convenient, and computationally attractive.

In this work, we show the existence of traveling wave solution of a diffusive predator-prey model with Holling type III functional response. The analysis is based on Wazewski's principle in the four-dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator-prey system under the moving coordinates.

In this paper, we propose a new non-linear partial differential equation to de-scribe the heat transfer problems at the extreme excess temperatures. Its exact traveling wave solutions are obtained by using Cornejo-Perez and Rosu method.

In this work, the modified simple equation (MSE) method is applied to some class of nonlinear PDEs, namely, a system of nonlinear PDEs, a (2 + 1)-dimensional nonlinear model generated by the Jaulent–Miodek hierarchy, and a generalized KdV equation with two power nonlinearities. As a result, exact traveling wave solutions involving parameters have been obtained for the considered nonlinear equations in a concise manner. When these parameters are chosen as special values, the solitary wave solutions are derived. It is shown that the proposed technique provides a more powerful mathematical tool for constructing exact solutions for a broad variety of nonlinear PDEs in mathematical physics.

Submitted by Maria Beatriz Vieira (mbeatriz.vieira@gmail.com) on 2019-08-14T18:09:13Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) dissertacao_daiane_frighetto.pdf: 1998663 bytes, checksum: 3c4c4da4338a7a0ae9567ed95e9a793b (MD5) Made available in DSpace on 2019-08-15T11:48:39Z (GMT). No. of bitstreams: 2 dissertacao_daiane_frighetto.pdf: 1998663 bytes, checksum: 3c4c4da4338a7a0ae9567ed95e9a793b (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-03-01 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES A utilização de modelos populacionais vem crescendo nas ciências agrárias e biológicas. Um dos principais interesses no estudo desses modelos é avaliar qualitativamente e quantitativamente o impacto das interações entre diferentes populações com o seu habitat ou entre elas, como: átomos ou moléculas, neurônios, bactérias, plantas, animais e até mesmo pragas ou indivíduos infectados por exemplo. Os modelos matemáticos descrevem o crescimento dessas populações com o propósito de fazer previsões, auxiliando na tomada de decisões caso haja necessidade da aplicação de controle nesses sistemas biológicos. Neste trabalho, defende-se a utilização do controle biológico, que consiste na regulação do crescimento da população através da intervenção humana ou por inimigos naturais, podendo ser muito eficiente no controle de pragas e ao mesmo tempo diminui a utilização de agrotóxicos, se tornando menos invasivo à natureza. Tem-se como objetivo estudar dois modelos populacionais de reação-difusão: a equação de Fisher-Kolmogorov para uma espécie, e o sistema de Lokta-Volterra modificado com difusão para duas espécies, ambos unidimensionais, sem e com a aplicação do controle. Esses modelos descrevem a dispersão de indivíduos em que o crescimento da população ocorre na forma de frente de onda, avançando em uma velocidade constante para as regiões vazias até atingir a capacidade de suporte do meio, assim, admitindo solução do tipo onda viajante, o que permite a realização de uma análise qualitativa da solução através da teoria de estabilidade e o encontro de soluções aproximadas. A partir do estudo realizado em ambos os modelos sem a influência de reguladores, aplica-se a teoria de controle ótimo linear considerando sistemas biológicos que necessitam da introdução de estratégias de controle para atingirem o equilíbrio desejado ou estabilizarem abaixo dos níveis de danos econômicos e/ou biológicos. Esse procedimento, parte inovadora desse trabalho, constitui uma metodologia eficaz na aplicação de controle em sistemas biológicos, cujo comportamento é modelado pelas equações de reação-difusão, que será ilustrada através das simulações computacionais. The use of population models is increasing in agrarian and biological sciences. One of the main interests in the study of these models is to evaluate qualitatively and quantitatively the impact of interactions between different populations with their habitat or between them, such as: atoms or molecules, neurons, bacteria, plants, animals and even pests or infected individuals for example. Mathematical models describe the growth of these populations for the purpose of making predictions, contribuing in the decision making if there is need for the application of control in these biological systems. In this work, we defend the use of biological control, which consists in regulating population growth through human intervention or natural enemies, being very efficient in pest control and at the same time reducing the use of pesticides, becoming less invasive to nature. The aim of this work is to study two population reaction-diffusion models: the Fisher-Kolmogorov equation for a species, and the modified Lokta-Volterra system with diffusion for two species, both in one dimension, with and without control. These models describe the dispersion of individuals in which the population growth occurs in the form of a wavefront, advancing at a constant velocity to the empty regions until reaching the supporting capacity of the medium, thus, admitting a traveling wave type solution, which allows a qualitative analysis of the solution through the theory of stability and the finding of approximate solutions. From the study carried out on both models without the influence of regulators, the theory of linear optimal control is applied considering biological systems that need the introduction of control strategies to achieve the wanted balance or stabilize below economic and/or biological damages. This procedure, an innovative part of this work, constitutes an effective methodology in the application of control in biological systems, whose behavior is modeled by the reaction-diffusion equation, which will be illustrated through computational simulations.

In this paper, we present the fractal complex transform via a local fractional derivative. The traveling wave solutions for the fractal Korteweg-de Vries equations within local fractional derivative are obtained based on the special functions defined on Cantor sets. The technology is a powerful tool for solving the local fractional non-linear partial differential equations.

Various motions of camphor boats in the water channel exhibit both a homogeneous and an inhomogeneous state, depending on the number of boats, when unidirectional motion along an annular water channel can be observed even with only one camphor boat. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Since the experimental results described above are thought of as a kind of bifurcation phenomena, we would like to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too difficult to analyze even if the number of camphor boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system.

The bifurcations of traveling wave solutions of a generalized Dullin-Gottwald-Holm equation ut−α2uxxt+2ωux+βumux+γuxxx = α2(2uxuxx+uuxxx) is studied by using the method of planar dynamical systems. Different kinds of traveling wave solutions, such as the solitary wave solution, the peakon wave solution and the periodic cusp wave solution are found to exist under certain parameter conditions. Results show that types of bounded traveling wave solutions are kept in the generalized Dullin-Gottwald-Holm equation.

We use the bifurcation method of dynamical systems to study the exact traveling wave solutions of the generalized Davey-Stewartson equation. The phase portraits are given. The bifurcation analysis will also be carried out. This analysis leads to several different solutions to this equation.

The existence of traveling waves is established for a diffusive SIV system with constant total population. The approach used is the geometric singular perturbation method. The same result is suitable to another SIV system with exponential input.

Kinetics of crystal growth in undercooled melts is analyzed by methods of theoretical modeling. Special attention is paid to rapid growth regimes occurring at deep undercoolings at which non-linearity in crystal velocity appears. A traveling wave solution of the phase field model (PFM) derived from the fast transitions theory is used for a quantitative description of the crystal growth kinetics. The “velocity – undercooling” relationship predicted by the traveling wave solution is compared with the data of molecular dynamics simulation (MDS) which were obtained for the crystal-liquid interfaces growing in the 〈 100〉-direction in the Ni50Al50 alloy melt.

The basic equations describing the nonlinear electron-acoustic waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions, in the long-wave limit, are re-examined through the use of the modified PLK method. Introducing the concept of strained coordinates and expanding the field variables into a power series of the smallness parameter epsilon, a set of evolution equations is obtained for various order terms in the perturbation expansion. The evolution equation for the lowest order term in the perturbation expansion is characterized by the conventional modified Korteweg-deVries (mKdV) equation, whereas the evolution equations for the higher order terms in the expansion are described by the degenerate(linearized) mKdV equation. By studying the localized traveling wave solution to the evolution equations, the strained coordinate for this order is determined so as to remove possible secularities that might occur in the solution. It is observed that the coefficient of the strained coordinate for this order corresponds to the correction term in the wave speed. The numerical results reveal that the contribution of second order term to the wave amplitude is about 20 %, which cannot be ignored. Publisher's Version

In the present paper, new analytical solutions for the fractional Vakhnenko–Parkes (VP) equation in the sense of the conformable derivative are obtained using the $$\exp (-\phi (\xi ))$$ expansion method. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Simulation of the obtained solutions are given at the end of the paper.