In this paper, we consider the (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) system of equations. This system of equations is a generalization of the cmKdV equation in the (2+1)-dimension and has great significance in the fields of applied magnetism and nanophysics. On the basis of the Lax pair, infinitely many conservation laws are obtained. In addition, the multi-waves, homo clinic breather, rational, and interactions solutions of this equation are derived with the aid of logarithmic transformation and symbolic computation. For the suitable value of parameters, the 3D surfaces of obtained solutions have been plotted using Mathematica.
In this paper, a nonlinear eigenvalue problem consisting of a nonlinear Sturm-Liouville equation -y ''- q(x)y =lambda q(-1)(x)y (R) with Dirichlet boundary conditions on the interval (-1/2, 1/2) is investigated, where lambda > 0 is the eigenparameter. We provide a simple scheme to obtain the asymptotic behavior of L-& ell;-bifurcation curve lambda = lambda(& ell;)(gamma) as gamma -> 0, where gamma =||y(lambda)||& ell;, & ell;>= 1, and y(lambda) is the solution of Dirichlet problem associated with lambda.
This paper examines the solutions of nonlinear higher-order singular Emden-Fowler type equations arising in various physical models. Generally, it becomes difficult to obtain the solution near the point of singularity. To overcome this problem, an iterative technique is introduced that depends on the variational iteration method (VIM) and the homotopy perturbation method (HPM). Such a technique generates the solution in terms of a series, which is highly practical from computing perspective. An equivalent recursive integral representation (involving Lagrange's multiplier) for the higher-order nonlinear singular Emden-Fowler type (SEFT) equations with initial conditions (ICs) is established with the support of the variational iteration method (VIM). Making use of the concept of homotopy, a system of integral equations is established, which helps to deal with nonlinearity. Some numerical examples are studied through the proposed iterative technique to show the applicability and efficiency of the technique.
In this article, we present a highly accurate technique for the numerical solution of the variable-order time-fractional Burgers-Huxley equation. The original equation is first discretized in the temporal and spatial directions. The third-order weighted-shifted Gru & uml;nwald-Letnikov and the fourth-order compact finite difference methods are used. We then formulate a nonlinear system of algebraic equations using the fully discretized version of the problem. The derived nonlinear system is solved by utilizing an iterative algorithm. The analysis of solvability, stability, and convergence of the method is also addressed. The method achieves a convergence rate of four in the spatial direction and three in the temporal direction. Moreover, it is a low-cost computational method and easy to implement. Finally, various illustrative examples are solved to verify the accuracy of the proposed method.
The need for accurate solutions to mathematical models, particularly for linear and nonlinear higher-order initial value problems, is essential across various scientific and engineering fields. Traditional methods often face challenges with stability and precision, especially in non-linear cases, prompting the development of advanced numerical techniques. This study introduces a two-step overlapping adaptive step-size multi-derivative hybrid block method to address these challenges in solving higher-order initial value problems. The method incorporates overlapping elements, using the second-to-last intra-step point from the previous step within each integration block to enhance accuracy. The method uses error estimation and selects an appropriate step-size, ensuring the desired accuracy without wasting computational resources or introducing unnecessary errors. The non-linear initial value problems are efficiently linearized using a modified-Picard iteration. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed method, and its performance is compared against a similar non-overlapping method as well as other methods reported in the literature.
In this study, we implicitly solve the generalized distributed-order time-fractional Black-Scholes equation. We employ finite differences to approximate the time derivatives and cubic B-spline quasi-interpolation for the spatial derivatives. The error analysis of the presented method is investigated. The algorithm of this method is also presented, which shows the simplicity of implementing the method to solve the generalized distributed-order time-fractional Black-Scholes equation. Numerical results demonstrate the method's convergence rate and accuracy.
In this paper, new analytical solutions of nonlinear fractional Wu-Zhang system are determined with the aid of two analytical approaches, that is, generalized projective Riccati equation method and Sardar sub-equation method via conformable derivative. The system describes (1 + 1)-dimensional dispersive long wave in two horizontal directions on shallow waters. Some new solitary wave solutions are demonstrated by the means of computer softwares maple or mathematica. The obtained results reveals that the proposed method is very efficacious and straightforward in the determination of the solution for the nonlinear fractional Wu-Zhang system.
In this paper, the power series method is applied to the fractional Lotka-Volterra equation, one of the most famous competition models in demography and economics. We obtain some power series solutions of the governing equation and prove their convergence. In addition, we analyze the various types of competitive roles depicted by this model through the truncated graphs of these power series solutions. From the graphs, we can find that the fractional order affects the speed of population growth or decrease, and this effect can be seen as continuous with respect to the order.
This paper focuses on optimizing the investment value function by incorporating jump risk using the Merton Jump-Diffusion (MJD) model. Our main goal is to determine the optimal dynamic asset allocation strategy to maximize expected utility. We derive the governing nonlinear Hamilton-Jacobi-Bellman (HJB) equation and employ a linearized generalized Newton method, which generates an iterative sequence for the optimal control. The theoretical convergence of this sequence was rigorously established using the Contraction Mapping Theorem, confirming the method's strong stability and reliability. Applying the model to real Go ogle stock data, which exhibit significant jump risks, we derived an optimal investment ratio (pi & lowast;) that suggests a notably aggressive allocation to the risky asset. This optimal strategy provides a direct, actionable benchmark for investors. Crucially, the derived dynamic control law functions as a powerful tool for investment management firms, enabling them to proactively adjust capital allocation strategies in response to potential future jump risk scenarios.
A mathematical and computational study of the impact of rotation on the thermal convection of partially-ionized plasma has been explored using both linear and nonlinear analyses. The method of normal mode analysis has been used to study the linear analysis whereas, for nonlinear analysis, we have used the generalized energy method. For numerical analysis, we have employed the Galerkin method. It has been found that the Rayleigh number for nonlinear analysis is the same as stationary convection. Hence, we concluded that there is no sub-critical region and the system is globally stable. The effect of collision plays an important role in the energy decay analysis. It has also been observed that for stationary convection, the collisional frequency has no impact on stability, whereas rotation stabilizes the system. The effect of various parameters has also been discussed for oscillatory convection. The stability characteristics for different bounding surfaces are examined. For low rotation rates, partially ionized plasma confined between rigid-rigid boundaries is the most stable configuration; however, at high rotation rates, the free-free bounding surfaces yield the greatest stability.
Kadomtsev-Petviashvili (KP) equation is an important (2+1)-dimensional nonlinear PDE which has not only multi-solitons but also has complete integrability. In order to describe the long waves that propagation weakly dispersive in the direction of additional spatial variable y, Kadomstev and Petviashili formulated this model. In the literature, many researchers are interested to propose and work on higher order nonlinear PDEs possessing multi-solitons. Two powerful methods employed by researchers are Hirota's method to obtain multi-solitons and tanh-coth method to obtain single-soliton solutions. In our work, a tenth-order generalization of the KP equation is derived and using Hirota's method, its multi-solitons are worked out. Furthermore, the derived equation is also treated with the tanh method. This article emphasizes few bounded solutions to the equation in context. The main aim of this paper is to demonstrate the generalization of the K-P equation using Hirota operators and to study corresponding multi-solitons. Finally, some open problems related to the proposed tenth-order KP equation are discussed.
The goal of the current study is to offer a novel method for numerically solving coupled 1D and 2D nonlinear Schro & uml;dinger equations. To discretize the spatial partial derivative, we applied the MCNUAH B-spline DQM. The SSP-RK43 technique is used to solve the reduced system of ODEs. Via the matrix method, the stability of the proposed method is investigated, and it is found to be stable. Four experiments are used to confirm the efficiency of the suggested scheme, and data from the literature are compared throughout. It is clear that the obtained results are satisfactory and in strong accord with preceding results. This approach yields superior outcomes and is effective, straightforward, and reasonably simple to use. The graphical abstract is provided as per Figure 1.
Naval hydrodynamics fundamentally depends on a detailed understanding of the boundary layers forming around a ship's hull, which generate resistance to advancement. Accurately modeling these layers is critical for calculating hydrodynamic resistance and estimating the propulsion power needed to achieve the desired speed specified by the shipowner. Traditionally, the velocity distribution within the boundary layer is described by the Blasius equation, a nonlinear third-order differential equation commonly solved using the Runge-Kutta numerical method, renowned for its accuracy. This study proposes a novel direct and explicit approach to solving the Blasius equation around a ship's hull, leveraging a derivative approximation technique implemented with MATLAB to obtain numerical results. By employing sufficiently small step sizes, the method produces highly accurate results that can serve as a benchmark for evaluating the precision of other numerical techniques applied in ship design. The proposed derivative approximation method provides a simple yet robust tool for solving complex differential equations, demonstrating its potential as an effective alternative for tackling problems similar to the Blasius equation in naval engineering applications.
In this study, a two-point composite block method based on the backward differentiation formula (CBBDF) is introduced to solve stiff ordinary differential equations. The CBBDF method incorporates an additional intermediate point among the interpolating points, developed in two stages: the first stage employs the Euler's method as a fundamental building block, while the second stage utilizes CBBDF of order three. A key distinction of the method with the classical block method is the introduction of an independent parameter gamma, which eliminates the need for an external startup calculation, while maintaining the accuracy and stability of numerical solutions. The theoretical analysis verifies that the proposed method is convergent and A-stable. It fulfills the essential properties of consistency and zero-stability, and it lies within the A-stability region. To demonstrate the effectiveness of the proposed approach, several stiff initial value problems of linear and non-linear are solved. For validation, the results are compared with existing literature. While approximating the solution at multiple points simultaneously, the composite block method offers the ability to use larger step sizes for solution approximation. The CBBDF method shows promising results, achieving a reliable degree of accuracy as indicated by its maximum error and average error measurements.
In this article, the propagation of modulated waves in one and two dimensional systems are analyzed by investigating the improved Eckhaus models analytically. Along with additional dimensions, dissipative factors, nonlo cal effects, and higher-order nonlinear elements, the enhanced Eckhaus equation expands the original Eckhaus equation. The investigation of the governing models' optical soliton solutions, including periodic, dark, brilliant, and singular solitons, is the focus of this article. This is done by obtaining a novel optical solution using the tanh-coth approach. Another type that incorporates nonlinearity and modulation effects in both spatial dimensions, and includes an extra spatial dimension, is the (2 + 1)-dimensional enhanced Eckhaus model. These equations are effective resources for examining a wide range of one- and two-dimensional system physical phenomena, including pattern generation, wave interaction, and soliton dynamics. Analyzing these equations can be challenging due to their higher dimensionality and nonlinear nature and numerical methods are often used to obtain solutions for specific cases or conditions. Consequently, trigonometric function solutions, hyperbolic function output and exponential functions solution with Independent parameters are acquired.3D and 2D contour plots of some solutions of the nonlinear model are specified. These governing equations have some applications in domains like nonlinear optics, condensed matter physics and fluid dynamics.
This paper introduces a monotonic weighted compact finite difference scheme (WC-FDM) designed to solve the non-linear one-dimensional steady advection-diffusion equation (ADE). The WC-FDM scheme is validated against the analytical solution and is adaptable to accommodate both uniform and non-uniform grid spacing. Criteria for selecting weights have been developed to ensure scheme monotonicity. Computational performance is benchmarked against other numerical schemes. Numerical analyses reveal that the WC-FDM accurately solves the non-linear steady ADE for both uniform and non-uniform grid spacing scenarios without introducing spurious oscillations. The proposed weight criteria maintain the monotonicity of the WC-FDM scheme resulting the computational stability regardless of the advection-dominance level and grid spacing uniformity.
The generalized solvability of a nonlinear optimal control for thermal and diffusion processes in a mixed inverse problem for a Barenblatt-Zheltov-Kochina differential equation with Hilfer fractional operator is studied. The inverse problem is considered with spectral and intermediate conditions. Eigenvalues, eigenfunctions, and associated functions of the spectral problem are found and the corresponding adjoint problem is solved. Countable systems of fractional order differential equations with final value conditions are obtained. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to solve a complicated nonlinear functional-integral equation, and the process of solving consists of solving separately taken two nonlinear functional-integral equations. Nonlinear functional integral equations are solved by the method of successive approximations and the unique solvability of these equations is proved by the method of contracting mapping. Approximate calculations for the optimal control function, the redefinition function, and the state function of the controlled process are obtained. The absolute and uniform convergence of the obtained Fourier series are proved.
The article focuses on investigating Lie symmetry analysis of the time-fractional Zeldovich-Frank-Kamenetskii equation with Riemann-Liouville derivative. The fractional reaction-diffusion equation describes how planar laminar premixed flames spread in combustion theory. The use of the Lie method is also illustrated to obtain Lie symmetry generators, symmetry reduction solutions, invariant properties, and conservation laws. Furthermore, we convert the time-fractional Zeldovich-Frank-Kamenetskii equation to a nonlinear fractional ordinary differential equation (ODE) with Erdelyi-Kober derivative using its Lie point symmetries. This decreased fractional ODE is investigated by explicit power series. In addition, some figures for the obtained explicit solution are presented.
This paper studies the dynamics of a SAIR mathematical model that describes the interaction among susceptible, asymptomatic, symptomatic, and recovered individuals. Two general incidence functions describing the infection caused by the asymptomatic and symptomatic individuals are introduced. We also take into account a temporary immunity, that is, a proportion of the recovered individuals becomes susceptible again. The basic reproduction number R0 depends on the general incidence functions. The local and global asymptotical stability for each equilibrium will depend on the basic reproduction number R0. In precise terms, the disease-free equilibrium is locally and globally asymptotically stable when R-0 < 1, while the endemic equilibrium is locally and globally asymptotic stable when R-0 > 1. The numerical simulation is performed for different incidence rate cases, such as bilinear, Beddington-DeAngelis, Crowley-Martin, and non-monotonic incidence rate functions. The simulation results are found to agree with the theoretical endings.
In this study, we investigate the significance of fractional kinetic equations in emerging a wide range of problems in science and engineering. Specifically, we derive a fractional kinetic equation solution involving the incomplete aleph function using a well-established integral transform technique. To illustrate the impact of the fractional integral operator's order on reaction rates, we present several graphical results, highlighting the influence of fractional calculus on the system's dynamics.
