scholarly journals Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zengji Du ◽  
Xiaojie Lin ◽  
Yulin Ren
Keyword(s):  
Lower Bounds ◽  
Solitary Waves ◽  
Wave Speed ◽  
Burgers Equation ◽  
Systems Approach ◽  
Upper And Lower Bounds ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Periodic Waves ◽  
Small Positive Parameter

<p style='text-indent:20px;'>This paper discusses the existence of solitary waves and periodic waves for a generalized (2+1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation with small damping and a weak local delay convolution kernel by using the dynamical systems approach, specifically based on geometric singular perturbation theory and invariant manifold theory. Moreover, the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. In addition, the upper and lower bounds and monotonicity of the period <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> of traveling wave when the small positive parameter <inline-formula><tex-math id="M2">\begin{document}$ \tau\rightarrow 0 $\end{document}</tex-math></inline-formula> are also obtained. Perhaps this paper is the first discussion on the solitary waves and periodic waves for the delayed KP-MEW-Burgers equations and the Abelian integral theory may be the first application to the study of the (2+1)-dimensional equation.</p>

scholarly journals EXISTENCE OF SOLITARY WAVES AND PERIODIC WAVES TO A PERTURBED GENERALIZED KDV EQUATION

2014 ◽  
Vol 19 (4) ◽  
pp. 537-555 ◽  
Author(s):  
Weifang Yan ◽  
Zhengrong Liu ◽  
Yong Liang
Keyword(s):  
Perturbation Theory ◽  
Solitary Waves ◽  
Wave Speed ◽  
Kdv Equation ◽  
Upper And Lower Bounds ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Periodic Waves ◽  
Regular Perturbation ◽  
Generalized Kdv Equation

In this paper, the existence of solitary waves and periodic waves to a perturbed generalized KdV equation is established by applying the geometric singular perturbation theory and the regular perturbation analysis for a Hamiltonian system. Moreover, upper and lower bounds of the limit wave speed are obtained. Some previous results are extended.

On the existence of traveling wave solutions and upper and lower bounds for some Fisher–Kolmogorov type equations

2014 ◽  
Vol 07 (05) ◽  
pp. 1450050 ◽  
Author(s):  
Juan Belmonte-Beitia
Keyword(s):  
Mathematical Biology ◽  
Dynamical Systems ◽  
Lower Bounds ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Systems Approach ◽  
Traveling Wave Solutions ◽  
Upper And Lower Bounds ◽  
Kolmogorov Type ◽  
Wave Solutions

In this paper, we use a dynamical systems approach to prove the existence of traveling waves solutions for the Fisher–Kolmogorov density-dependent equation. Moreover, we prove the existence of upper and lower bounds for these traveling wave solutions found previously. Finally, we present a particular example which has several applications in the mathematical biology field.

scholarly journals Minimum wave speeds in monostable reaction–diffusion equations: sharp bounds by polynomial optimization

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200450
Author(s):  
Jason J. Bramburger ◽  
David Goluskin
Keyword(s):  
Lower Bounds ◽  
Wave Speed ◽  
Polynomial Optimization ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Upper And Lower Bounds ◽  
Wave Speeds ◽  
Minimum Wave Speed

Many monostable reaction–diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction–diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction–diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction–diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.

scholarly journals Propagating fronts for a viscous Hamer-type system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Giada Cianfarani Carnevale ◽  
Corrado Lattanzio ◽  
Corrado Mascia
Keyword(s):  
Perturbation Theory ◽  
Elliptic Equation ◽  
Burgers Equation ◽  
Type System ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Radiation Hydrodynamics ◽  
One Dimensional ◽  
Small Viscosity ◽  
Viscous Conservation Law

<p style='text-indent:20px;'>Motivated by radiation hydrodynamics, we analyse a <inline-formula><tex-math id="M1">\begin{document}$ 2\times2 $\end{document}</tex-math></inline-formula> system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named <b>viscous Hamer-type system</b>. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity –usually called <i>sub-shock</i>– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on <i>Geometric Singular Perturbation Theory</i> (GSPT) as introduced in the pioneering work of Fenichel [<xref ref-type="bibr" rid="b5">5</xref>] and subsequently developed by Szmolyan [<xref ref-type="bibr" rid="b21">21</xref>]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.</p>

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

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