scholarly journals Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1440
Author(s):  
Mostafa M. A. Khater ◽  
Aliaa Mahfooz Alabdali
Keyword(s):  
Population Genetics ◽  
Liquid Crystals ◽  
Analytical Solutions ◽  
Evolution Equations ◽  
Domain Walls ◽  
Numerical Solutions ◽  
Nematic Liquid Crystals ◽  
Reaction Diffusion ◽  
Nonlinear Evolution Equations ◽  
Absolute Value

The analytical and numerical solutions of the (2+1) dimensional, Fisher-Kolmogorov-Petrovskii-Piskunov ((2+1) D-Fisher-KPP) model are investigated by employing the modified direct algebraic (MDA), modified Kudryashov (MKud.), and trigonometric-quantic B-spline (TQBS) schemes. This model, which arises in population genetics and nematic liquid crystals, describes the reaction–diffusion system by traveling waves in population genetics and the propagation of domain walls, pattern formation in bi-stable systems, and nematic liquid crystals. Many novel analytical solutions are constructed. These solutions are used to evaluate the requested numerical technique’s conditions. The numerical solutions of the considered model are studied, and the absolute value of error between analytical and numerical is calculated to demonstrate the matching between both solutions. Some figures are represented to explain the obtained analytical solutions and the match between analytical and numerical results. The used schemes’ performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations.

New Travelling Wave Solutions of Burgers Equation with Finite Transport Memory

2010 ◽  
Vol 65 (8-9) ◽  
pp. 633-640 ◽  
Author(s):  
Rathinasamy Sakthivel ◽  
Changbum Chun ◽  
Jonu Lee
Keyword(s):  
Evolution Equations ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Diffusion Processes ◽  
Initial Conditions ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions ◽  
Wide Range ◽  
Finite Memory

The nonlinear evolution equations with finite memory have a wide range of applications in science and engineering. The Burgers equation with finite memory transport (time-delayed) describes convection-diffusion processes. In this paper, we establish the new solitary wave solutions for the time-delayed Burgers equation. The extended tanh method and the exp-function method have been employed to reveal these new solutions. Further, we have calculated the numerical solutions of the time-delayed Burgers equation with initial conditions by using the homotopy perturbation method (HPM). Our results show that the extended tanh and exp-function methods are very effective in finding exact solutions of the considered problem and HPM is very powerful in finding numerical solutions with good accuracy for nonlinear partial differential equations without any need of transformation or perturbation

scholarly journals Exact and Numerical Solitary Wave Structures to the Variant Boussinesq System

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1473 ◽  
Author(s):  
Abdulghani Alharbi ◽  
Mohammed B. Almatrafi
Keyword(s):  
Solitary Wave ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Numerical Technique ◽  
Method Of Lines ◽  
Nonlinear Evolution Equations ◽  
Boussinesq System ◽  
Solitary Wave Solutions ◽  
The Method Of Lines

Solutions such as symmetric, periodic, and solitary wave solutions play a significant role in the field of partial differential equations (PDEs), and they can be utilized to explain several phenomena in physics and engineering. Therefore, constructing such solutions is significantly essential. This article concentrates on employing the improved exp(−ϕ(η))-expansion approach and the method of lines on the variant Boussinesq system to establish its exact and numerical solutions. Novel solutions based on the solitary wave structures are obtained. We present a comprehensible comparison between the accomplished exact and numerical results to testify the accuracy of the used numerical technique. Some 3D and 2D diagrams are sketched for some solutions. We also investigate the L2 error and the CPU time of the used numerical method. The used mathematical tools can be comfortably invoked to handle more nonlinear evolution equations.

scholarly journals Flow invariance for perturbed nonlinear evolution equations

1996 ◽  
Vol 1 (4) ◽  
pp. 417-433 ◽  
Author(s):  
Dieter Bothe
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Accretive Operator ◽  
Nonlinear Evolution Equations ◽  
Evolution System ◽  
Reaction Diffusion Systems ◽  
Closed Sets

LetXbe a real Banach space,J=[0,a]⊂R,A:D(A)⊂X→2X\ϕanm-accretive operator andf:J×X→Xcontinuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed setsK⊂Xfor the evolution systemu′+Au∍f(t,u)  on  J=[0,a]. More generally, we provide conditions under which this evolution system has mild solutions satisfying time-dependent constraintsu(t)∈K(t)onJ. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of typeut=ΔΦ(u)+g(u)  in  (0,∞)×Ω,   Φ(u(t,⋅))|∂Ω=0,   u(0,⋅)=u0under certain assumptions on the setΩ⊂Rnthe functionΦ(u1,…,um)=(φ1(u1),…,φm(um))andg:R+m→Rm.

scholarly journals Dynamics of cylindrical domain walls in nematic liquid crystals

1997 ◽  
Vol 56 (2) ◽  
pp. 1784-1790 ◽  
Author(s):  
Joachim Stelzer ◽  
Henryk Arodź
Keyword(s):  
Liquid Crystals ◽  
Domain Walls ◽  
Nematic Liquid Crystals ◽  
Cylindrical Domain

scholarly journals Solving Some Linear and Nonlinear PDEs Using Laplace Adomian Decomposition Method

2018 ◽  
Vol 1 (2) ◽  
pp. 9-31
Author(s):  
Attaullah
Keyword(s):  
Decomposition Method ◽  
Evolution Equations ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Reaction Diffusion ◽  
Nonlinear Evolution Equations ◽  
Reaction Diffusion Equations ◽  
Nonlinear Pdes ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

In this paper, Laplace Adomian decomposition method (LADM) is applied to solve linear and nonlinear partial differential equations (PDEs). With the help of proposed method, we handle the approximated analytical solutions to some interesting classes of PDEs including nonlinear evolution equations, Cauchy reaction-diffusion equations and the Klien-Gordon equations.

scholarly journals Gradient theory of domain walls in thin, nematic liquid crystals films

2019 ◽  
Vol 22 (07) ◽  
pp. 1950063
Author(s):  
Marcel G. Clerc ◽  
Michał Kowalczyk ◽  
Panayotis Smyrnelis
Keyword(s):  
Phase Transition ◽  
Liquid Crystal ◽  
Liquid Crystals ◽  
External Field ◽  
Nematic Liquid Crystal ◽  
Domain Walls ◽  
Nematic Liquid Crystals ◽  
Gradient Theory ◽  
Crystal Sample ◽  
Bistable Region

In this paper, we describe domain walls appearing in a thin, nematic liquid crystal sample subject to an external field with intensity close to the Fréedericksz transition threshold. Using the gradient theory of the phase transition adapted to this situation, we show that depending on the parameters of the system, domain walls occur in the bistable region or at the border between the bistable and the monostable region.

Abundant breather and semi-analytical investigation: On high-frequency waves’ dynamics in the relaxation medium

2021 ◽  
pp. 2150372
Author(s):  
Mostafa M. A. Khater
Keyword(s):  
Analytical Solutions ◽  
High Frequency ◽  
Evolution Equations ◽  
Dynamical Behavior ◽  
Analytical Investigation ◽  
Nonlinear Evolution Equations ◽  
Research Papers ◽  
Ostrovsky Equation ◽  
High Frequency Waves ◽  
Published Research

This paper investigates the high-frequency waves’ dynamical behavior in the relaxation medium through two recent analytical schemes. This study depends on the Vakhnenko–Parkes (VP) equation that has been reduced from the well-known Ostrovsky equation. The modified Khater (MKhat) and the extended simplest equation (ESE) methods are used to handle the considered model. As a result, many novel solitary wave solutions have been obtained to construct the initial and boundary conditions. These conditions allow employing the variational iteration (VI) method to study the semi-analytical solutions of the considered model. The accuracy of solutions is explained along with showing the matching between analytical and semi-analytical solutions and comparing our obtained solutions with the previous results that have been obtained in published research papers. Moreover, the high-frequency waves’ behavior relaxation medium is illustrated through some distinct sketches. The methods’ performance shows their effectiveness, direct, easy, and consequential for studying many nonlinear evolution equations.

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