Computational Methods for Differential Equations
Volume:13 Issue: 1, Winter 2025

In this work, we established some exact solutions for the (2+1)-dimensional Zakharov-Kuznetsov, KdV, and K(2,2) equations which are considered based on the improved Exp-function method, by utilizing Maple software. We use the fractional derivatives with fractional complex transform. We obtained new periodic solitary wave solutions. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. Many other such types of nonlinear equations arising in fluid dynamics and nonlinear phenomena.

This research introduces a novel approach using artificial neural networks (ANNs) to tackle ordinary differential equations (ODEs) through an innovative technique called enhanced back-propagation (EBP). The ANNs adopted in this study, particularly multilayer perceptron neural networks (MLPNNs), are equipped with tunable parameters such as weights and biases. The utilization of MLPNNs with universal approximation capabilities proves to be advantageous for ODE problem solving. By leveraging the enhanced back-propagation algorithm, the network is fine-tuned to minimize errors during unsupervised learning sessions. To showcase the effectiveness of this method, a diverse set of initial value problems for ODEs are solved and the results are compared against analytical solutions and conventional techniques, demonstrating the superior performance of the proposed approach.

This study attempts to find approximate numerical solutions for a kind of second-order nonlinear differential problem subject to some Dirichlet and mixed-type nonlocal (specifically three-point) boundary conditions, appearing in various realistic physical phenomena, such as bridge design, control theory, thermal explosion, thermostat model, and the theory of elastic stability. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of nonlinear differential problems with nonlocal boundary conditions. Picard’s iterative technique and quasilinearization method are the basis for the proposed coupled iterative methodology. In order to convert nonlinear boundary value problems to linearized form, the quasilinearization approach (with convergence controller parameters) is implemented. Making use of Picard’s iteration method with the assistance of Green’s function, an equivalent integral representation for the linearized problems is derived. Discussion is also had over the proposed method’s convergence analysis. In order to determine its efficiency and effectiveness, the coupled iterative technique is tested on some numerical examples. Results are also compared with the existing techniques and documented (in terms of absolute errors) to validate the accuracy and precision of the proposed iterative technique.

The stable Gaussian radial basis function (RBF) interpolation is applied to solve the time and space-fractional Schrödinger equation (TSFSE) in one and two-dimensional cases. In this regard, the fractional derivatives of stable Gaussian radial basis function interpolants are obtained. By a method of lines, the computations of the TSFSE are converted to a coupled system of Caputo fractional ODEs. To solve the resulting system of ODEs, a high-order finite difference method is proposed, and the computations are reduced to a coupled system of nonlinear algebraic equations, in each time step. Numerical illustrations are performed to certify the ability and accuracy of the new method. Some comparisons are made with the results in other literature.

This study utilizes two robust methodologies to examine the precise solutions of the Dirac integrable system. The Homogeneous Balance Method (HB) is initially employed to generate an accurate solution. The system of equations for the quasi-solution is solved, where all the equations are of the same nature. The quasi-solution of the traveling wave results in the solitary wave solution of the system. The singular manifold method (SMM) is utilized following the Lie reduction of the Dirac system in order to search for the traveling wave solutions of the system. Both approaches demonstrate the existence of traveling wave solutions inside the system. The precise solutions of the Dirac system are shown in three-dimensional graphs. We have created solutions to the examined problem, including bright solutions, periodic soliton solutions, and complicated solutions.

In this work, we examine the existence and uniqueness(EU) of q-Exponential positive solution (q-EPS) of the hybrid q-fractional boundary value problem (q-FBVP).We prove the q-Exponential fixed point theorem (q-EFPT) with a new set $\rho_{h,e_{1}}$ in the Banach space E to check the EU of q-EPS of the q-FBVP. In the long run, an exemplum is given to show the correctness of our results.

In this study, the time-fractional Newell-Whitehead-Segel (NWS) equation and its different nonlinearity cases are investigated. Schemes obtained by the Newtonian linearization method are used to numerically solve different cases of the time-fractional Newell-Whitehead-Segel (NWS) equation. Stability and convergence conditions of the Newtonian linearization method have been determined for the related equation. The numerical results obtained as a result of the appropriate stability criteria are compared with the help of tables and graphs with exact solutions for different fractional values.

In this paper, an efficient high-order compact finite difference (HOCFD) scheme is introduced for solving generalized Lane-Emden equations. For nonlinear types, it is shown that a combined quasilinearization and HOCFD scheme gives excellent results, while a few quasilinear iterations are needed. Then the proposed method is developed for solving the system of linear and nonlinear Lane-Emden equations. Some numerical examples are provided, and the obtained results of the proposed method are then compared with previous well-established methods. The numerical experiments show the accuracy and efficiency of the proposed method.

COVID-19 was declared a pandemic on March 11, 2020, after the global cases and mortalities in more than 100 countries surpassed 100 000 and 3 000, respectively. Because of the role of isolation in disease spread and transmission, a system of differential equations were developed to analyse the effect of isolation on the dynamics of COVID-19. The validity of the model was confirmed by establishing the positivity and boundedness of its solutions. Equilibria analysis was conducted, and both zero and nonzero equilibria were obtained. The effective and basic reproductive ratios were also derived and used to analyze the stability of the equilibria. The disease-free equilibrium is stable both locally and globally if the reproduction number is less than one; otherwise, it is the disease-endemic equilibrium that is stable locally and globally. A numerical simulation was carried out to justify the theoretical results and to visualise the effects of various parameters on the dynamics of the disease. Results from the simulations indicated that COVID-19 incidence and prevalence depended majorly on the effective contact rate and per capita probability of detecting infection at the asymptomatic stage, respectively. The policy implication of the result is that disease surveillance and adequate testing are important to combat pandemics.

Using different algorithms to extract, describe, and match features requires knowing their capabilities and weaknesses in various applications. Therefore, it is a basic need to evaluate algorithms and understand their performance and characteristics in various applications. In this article, classical local feature extraction and description algorithms for large-scale satellite image matching are discussed. Eight algorithms, SIFT, SURF, MINEIGEN, MSER, HARRIS, FAST, BRISK, and KAZE, have been implemented, and the results of their evaluation and comparison have been presented on two types of satellite images. In previous studies, comparisons have been made between local feature algorithms for satellite image matching. However, the difference between the comparison of algorithms in this article and the previous comparisons is in the type of images used, which both reference and query images are large-scale, and the query image covers a small part of the reference image. The experiments were conducted in three criteria: time, repeatability, and accuracy. The results showed that the fastest algorithm was Surf, and in terms of repeatability and accuracy, Surf and Kaze got the first rank, respectively.

A two-dimensional coupled hygrothermoelastic medium boundary problem using Finite difference method is discussed in the present work. Explicit and Implicit finite difference schemes for this problem are formed. The solutions of these schemes are carried out using numerical methods of finite difference. These solutions are compared of and analyzed and exciting similarities were found as result.

Ahmad et al. (see [1]) presented a piecewise Lagrange interpolation method for solving tumor-immune interaction models with fractal fractional operators using a power law and exponential kernel. We suggest a convergence analysis for this method and we obtain the order of convergence. Of course, there are some mistakes in this numerical method that were corrected. Furthermore, Numerical illustrations are demonstrated to show the effectiveness of the corrected numerical method.

The authors consider Hammerstein-type integral equations for the purpose of obtaining new results on the uniqueness of solutions on an infinite interval. The approach used in the proofs is based on the technique called progressive contractions due to T. A. Burton. Here the authors apply the Burton’s method to a general Hammerstein type integral equation that also yields the existence of solutions. In most of the existing literature, investigators prove uniqueness of solutions of integral equations by applying some type of fixed point theorem which can be tedious and challenging, often patching together solutions on short intervals after making complicated translations. In this article, using the progressive contractions throughout three simple short steps, each of the three steps is an elementary contraction mapping on a short interval, we improve the technique due to T. A. Burton for a general Hammerstein type integral equation and obtain the uniqueness of solutions on an infinite interval. These are advantages of the used method to prove the uniqueness of solutions.

Vascular-related diseases have become increasingly significant as public health concerns. The analysis of blood vessels plays an important role in detecting and treating diseases. Extraction of vessels is a very important technique in vascular analysis. Magnetic Resonance Angiography (MRA) is a medical imaging technique used to visualize the blood vessels and vascular system in three-dimensional images. These images provide detailed information about the size and shape of the vessels, any narrowing or stenosis, as well as blood supply and circulation in the body. Tracing vessels from medical images is an essential step in the diagnosis and treatment of vascular-related diseases. Many different techniques and algorithms have been proposed for vessel extraction. In this paper, we present a vessel extraction method based on the Kernel density estimation (KDE). Numerical experiments on real 2D MRA images demonstrate that the presented method is very efficient. The effectiveness of the proposed method has been proven through comparative analysis with validated existing methods.

Nigeria is one of the most populated countries in West Africa and is in seventh position globally. The issue of terrorism has become a common problem in Nigeria, and the government has been applying local strategies to address the situation but has yet to produce good results. The challenges necessitate the effort in this paper to develop a new deterministic model to curb terrorism and insurgency through technology intelligence in Nigeria. This analysis indicates that Unmanned Aerial Vehicles (UAV) and the transmission rate per capita are the most sensitive parameters. Also pictured from the graphs in Figures 2, 3, and 4 were drones used to reduce the number of informants of both the terrorist and kidnapper individuals in Nigeria. Finally, this paper recommended the model adopted for controlling terrorism in Nigeria.

In this paper, the discretization method is developed by means of Mott-fractional Mott functions (MFM-Fs) for solving fractional partial integro-differential viscoelastic equations with weakly singular kernels. By taking into account the Riemann-Liouville fractional integral operator and operational matrix of integration, we convert the proposed problem to fractional partial integral equations with weakly singular kernels. It is necessary to mention that the operational matrices of integration are obtained with new numerical algorithms. These changes effectively affect the solution process and increase the accuracy of the proposed method. Besides, we investigate the error analysis of the approach. Finally, several examples are solved by applying the discretization method by combining MFM-Fs and the gained results are compared with the methods available in the literature.

In this article, the exact solutions for nonlinear Drinfeld-Sokolov (DS) and generalized Drinfeld-Sokolov (gDS) equations are established. The rational Exp-function method (EFM) is used to construct solitary and soliton solutions of nonlinear evolution equations. This method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. Also, exact solutions with solitons and periodic structures are obtained. The obtained results are not only presented numerically but are also accompanied by insightful physical interpretations, enhancing the understanding of the complex dynamics described by these mathematical models. The utilization of the rational EFM and the broad spectrum of obtained solutions contribute to the depth and significance of this research in the field of nonlinear wave equations.

In this work, we study a hypersurface immersed in specific types of cylindrically symmetric static space-times, then we identify the gauge fields of the Lagrangian that minimizes the area beside the Noether symmetries. We show that these symmetries are part of the Killing algebra of cylindrically symmetric static space-times. By using Noether’s theorem, we construct the conserved vector fields for the minimal hypersurface.

The generalized ( G′/G )-expansion method with the aid of Maple is proposed to seek exact solutions of nonlinear evolution equations. For finding exact solutions are expressed three types of solutions that include hyperbolic function solution, trigonometric function solution, and rational solution. The article studies the Zakharov–Kuznetsov (ZK) equation, the generalized ZK (gZK) equation, and the generalized forms of these equations. Exact solutions with traveling wave solutions of nonlinear evolution equations are obtained. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations.

Optical solitons are self-trapped light beams that maintain their shape and transverse dimension during propagation. This paper investigates the propagation of solitons in an optical material with a weak nonlocal media, modeled by a cubic-quintic-septimal nonlinearity. The dynamics of solitons in optical waveguides are described by the cubic nonlinear Schrödinger equation and its extensions. This equation model applies to both the spatial propagation of beams and the temporal propagation of pulses in a medium exhibiting cubic nonlinearity. The novelty of the paper lies in the application of the extended hyperbolic function method to derive soliton solutions in optical materials with weak nonlocal media in the form of the periodic, bright, kink, and singular type solitons. The obtained solutions provide explicit expressions for the behavior of optical waves in media. These results shed light on the dynamics of nonlinear waves in optical materials and contribute to a better understanding of soliton propagation. The findings contribute to a more comprehensive understanding of the role of nonlocal nonlinearity and time constants in soliton solutions. Our findings provide a better understanding of the dynamics of the nonlinear waves in optical media and have many application for the field of optical communication and signal processing. The role of nonlocal nonlinearity and time constant on soliton solutions is also discussed with the help of graphs.

In this current article, the well-known Neumann method for solving the time M-fractional Volterra integral equations of the second kind is developed. In the several theorems, existence and uniqueness of the solution and convergence of the proposed approach are also studied. The Neumann method for this class of the time M-fractional Volterra integral equations has been called the M-fractional Neumann method (MFNM). The results obtained demonstrate the efficiency of the proposed method for the time M-fractional Volterra integral equations. Several illustrative numerical examples have presented the ability and adequacy of the MFNM for a class of fractional integral equations.

This paper is concerned with a cross-diffusion prey-predator system in which the prey species is equipped with the group defense ability under the Neumann boundary conditions. The tendency of the predator to pursue the prey is expressed in the cross-diffusion coefficient, which can be positive, zero, or negative. We first select the environmental protection of the prey population as a bifurcation parameter. Next, we discuss the Turing instability and the Hopf bifurcation analysis on the proposed cross-diffusion system. We show that the system without cross-diffusion is stable at the constant positive stationary solution but it becomes unstable when the cross-diffusion appears in the system. Furthermore, the stability of bifurcating periodic solutions and the direction of Hopf bifurcation are examined.

A unified explicit form for difference formulas to approximate fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivative with a desired order of accuracy at any nodal point in computational domain. It also gives Gr¨unwald type approximations for fractional derivatives with arbitrary order of approximation at any nodal point. Thus, this explicit form unifies approximations of both types of derivatives. Moreover, for classical derivatives, it also provides various finite difference formulas such as forward, backward, central, staggered, compact, non-compact, etc. Efficient computations of the coefficients of the difference formulas are also presented leading to automating the solution process of differential equations with a given higher order accuracy. Some basic applications are presented to demonstrate the usefulness of this unified formulation.

The problem of minimizing a function of three criteria maximum earliness, total of square completion times, and total lateness in a hierarchical (lexicographical) method is proposed in this article. On one machine, n independent tasks (jobs) must planned. It is always available starting at time zero and can only do the mono task (job) at a time period. Processing for the task (job) j (j = 1, 2, ..., nj) is necessary meantime the allotted positive implementation time $p_{tj}$ . For the problem of three criteria maximization earliness, a total of square completion times, and total lateness in a hierarchy instance, the access of limitation that which is the desired sequence is held out. The Generalized Least Deviation Method (GLDM) and a robust technique for analyzing historical data to project future trends are analyzed.

The standard model, which determines option pricing, is the well-known Black-Scholes formula. Heston in addition to Cox-Ingersoll-Ross which is called CIR, respectively, implemented the models of stochastic volatility and interest rate to the standard option pricing model. The cost of transaction, which the Black-Scholes method overlooked, is another crucial consideration that must be made when trading a service or production. It is acknowledged that by employing the log-normal stock diffusion hypothesis with constant volatility, the Black-Scholes model for option pricing departs from reality. The standard log-normal stock price distribution used in the Black-Scholes model is insufficient to account for the leaps that regularly emerge in the discontinuous swings of stock prices. A jump-diffusion model, which combines a jump process and a diffusion process is a type of mixed model in the Black-Scholes model belief. Merton developed a jump model as a modification of jump models to better describe purchasing and selling behavior. In this study, the Heston-Cox-Ingersoll-Ross (HCIR) model with transaction costs is solved using the alternating direction implicit (ADI) approach and the Monte Carlo simulation assuming the underlying asset adheres to the jump-diffusion case, then the outcomes are compared to the analytical solution. In addition, the consistency of the numerical method is proven for the model.