Accuracy Analysis on Solution of Initial Value Problems of Ordinary Differential Equations for Some Numerical Methods with Different Step Sizes

In this article, three numerical methods namely Euler’s, Modified Euler, and Runge-Kutta method have been discussed, to solve the initial value problem of ordinary differential equations. The main goal of this research paper is to find out the accurate results of the initial value problem (IVP) of ordinary differential equations (ODE) by applying the proposed methods. To achieve this goal, solutions of some IVPs of ODEs have been done with the different step sizes by using the proposed three methods, and solutions for each step size are analyzed very sharply. To ensure the accuracy of the proposed methods and to determine the accurate results, numerical solutions are compared with the exact solutions. It is observed that numerical solutions are best fitted with exact solutions when the taken step size is very much small. Consequently, all the proposed three methods are quite efficient and accurate for solving the IVPs of ODEs. Error estimation plays a significant role in the establishment of a comparison among the proposed three methods. On the subject of accuracy and efficiency, comparison is successfully implemented among the proposed three methods

Complex Dynamics and Chaos Control of a Discrete-Time Predator-Prey Model

The objective of this study is to investigate the complexity of a discrete predator-prey system. The discretization is achieved using the piecewise constant argument method. The existence and stability of equilibrium points, as well as transcritical and Neimark–Sacker bifurcations, are all explored. Feedback and hybrid control methods are used to control the discrete system’s bifurcating and fluctuating behavior. To validate the theoretical conclusions, numerical simulations are performed. The findings of the study suggested that the discretization technique employed in this investigation preserves bifurcation and displays more effective dynamic consistency in comparison to the Euler method

Stable and effective traveling wave solutions to the non-linear fractional Gardner and Zakharov–Kuznetsov–Benjamin–Bona–Mahony equations

The space–time fractional Gardner and Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equations are used to explain the transmission of shallow water waves inside a water channel of uniform speed and constant depth. It moreover simulates water waves in harbors and shallow water, which is useful for oceanography. These equations are significant nonlinear model equations to illustrate numerous physical structures namely in marine and coastal science, water wave mechanics, control theory, plasma engineering, fluid flow, optical fibers, phenomena of fission and fusion, and so on. In this study, we employed the extended tanh-function approach to generate some fresh and further general closed-form traveling waves solutions of those equations in the lite of conformal derivatives. The achieve results are more applicable to resolve the above mentioned phenomena properly. The fractional differential transform simplifies generate ordinary differential equations from fractional order differential equations. We discovered many types of solutions using the maple, including solitons, kink types, bell types, and other types of solutions that are illustrated using 3D and contour plots. It is important to note that all derived solutions are checked for accuracy by being directly replaced with the original equation. We proposed that the technique be revised to be more realistic, effective, and trustworthy and that we explore more generalized exact results of traveling waves, like solitary wave solutions

New Explicit Solutions to the Fractional-Order Burgers’ Equation

The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables G′/G,1/G-expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of tanh, sech, sinh, rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results

Solitary wave solution to the space–time fractional modified Equal Width equation in plasma and optical fiber systems

Nonlinear fractional evolution equations play a crucial role in characterizing assorted complex nonlinear phenomena observed in different scientific fields, including plasma physics, quantum mechanics, elastic media, nonlinear optics, surface water waves, nonlinear dynamics, molecular biology, and some other domains. In this study, we investigate diverse solitary wave solutions to the space–time fractional modified Equal Width equation, focusing on their significance for wave propagation behaviors in plasma and optical fiber systems. Two effective approaches, namely the improved Bernoulli sub-equation function and the new generalized (G′/G)-expansion methods are exploited. A fractional wave transformation technique is employed to convert the fractional-order equation into an ordinary differential equation. The solitary wave solutions obtained in terms of exponential, hyperbolic, rational, and trigonometric functions, have wide applications in various nonlinear phenomena. The 3D, spherical, contour, and vector plots are presented to elucidate the physical implications of the obtained solutions for specific parameter values. The soliton structures reveal various wave types, including bell-shaped, multi-soliton, kink, single soliton, anti-bell-shaped, periodic soliton, compacton, and other types for definite values of the free parameters. We compare the attained solutions with the solutions available in the literature to demonstrate the feasibility of the solutions. It is observed that both approaches are efficient, straightforward, and significant for investigating diverse nonlinear models that modulate complex phenomena

A study of the wave dynamics of the space–time fractional nonlinear evolution equations of beta derivative using the improved Bernoulli sub-equation function approach

Abstract The space–time fractional nonlinear Klein-Gordon and modified regularized long-wave equations explain the dynamics of spinless ions and relativistic electrons in atom theory, long-wave dynamics in the ocean, like tsunamis and tidal waves, shallow water waves in coastal sea areas, and also modeling several nonlinear optical phenomena. In this study, the improved Bernoulli sub-equation function method has been used to generate some new and more universal closed-form traveling wave solutions of those equations in the sense of beta-derivative. Using the fractional complex wave transformation, the equations are converted into nonlinear differential equations. The achieved outcomes are further inclusive of successfully dealing with the aforementioned models. Some projecting solitons waveforms, including, kink, singular soliton, bell shape, anti-bell shape, and other types of solutions are displayed through a three-dimensional plotline, a plot of contour, and a 2D plot for definite parametric values. It is significant to note that all obtained solutions are verified as accurate by substituting the original equation in each case using the computational software, Maple. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed technique is effective, computationally attractive, and trustworthy to establish more generalized wave solutions

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique.

Nonlinear fractional partial differential equations are highly applicable for representing a wide variety of features in engineering and research, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. In this research, we chose to construct some new closed form solutions of traveling wave of fractional order nonlinear coupled type Boussinesq-Burger (BB) and coupled type Boussinesq equations. In beachside ocean and coastal engineering, the suggested equations are frequently used to explain the spread of shallow-water waves, depict the propagation of waves through dissipative and nonlinear media, and appears during the investigation of the flow of fluid within a dynamic system. The subsidiary extended tanh-function technique for the suggested equations is solved for achieve new results by conformable derivatives. The fractional order differential transform was used to simplify the solution process by converting fractional differential equations to ordinary type differential equations by using the mentioned method. Using this technique, some applicable wave forms of solitons like bell type, kink type, singular kink, multiple kink, periodic wave, and many other types solution were accomplished, and we express our achieve solutions by 3D, contour, list point, and vector plots by using mathematical software such as MATHEMATICA to express the physical sketch much more clearly. Moreover, we assured that the suggested technique is more reliable, pragmatic, and dependable, that also explore more general exact solutions of close form traveling waves

Sharp Bounds of Kulli–Basava Indices in Generalized Form for k-Generalized Quasi Trees

Molecular descriptors are a basic tool in the spectral graph, molecular chemistry, and various other fields of mathematics and chemistry. Kulli–Basava KB indices were initiated for chemical applications of various substances in chemistry. For simple graph G, KB indices in generalized forms are KB1ϱG=∑gh∈EGSeg+Sehϱ and KB2ϱG=∑gh∈EGSeg.Sehϱ, where Seg=∑e∈NegdGe, and for edge e=g,h, the degree is dGe=dGg+dGh−2 and ϱ≠0 is any real number. The graph G is said to be a k−generalized quasi tree if for the vertex set Uk⊂G having Uk=k, G−Uk is a tree and for Uk−1⊂VG having Uk−1=k−1, G−Uk−1 is not a tree. In this research work, we have successfully investigated sharp bounds of generalized KB indices for k-generalized quasi trees where ϱ≥1. Chemical applications of the generalized form are also studied for alkane isomers with scatter diagrams and residuals