Search Results (39)

This manuscript aims to explore localized waves for the nonlinear partial differential equation referred to as the $(1+1)$-dimensional generalized Kundu–Eckhaus equation with an additional dispersion term that describes the propagation of the ultra-short femtosecond pulses in an optical fiber. This research delves deep into the characteristics, behaviors, and localized waves of the $(1+1)$-dimensional generalized Kundu–Eckhaus equation. We utilize the multivariate generalized exponential rational integral function method (MGERIFM) to derive localized waves, examining their properties, including propagation behaviors and interactions. Motivated by the generalized exponential rational integral function method, it proves to be a powerful tool for finding solutions involving the exponential, trigonometric, and hyperbolic functions. The solutions we found using the MGERIF method have important applications in different scientific domains, including nonlinear optics, plasma physics, fluid dynamics, mathematical physics, and condensed matter physics. We apply the three-dimensional (3D) and contour plots to illuminate the physical significance of the derived solution, exploring the various parameter choices. The proposed approaches are significant and applicable to various nonlinear evolutionary equations used to model nonlinear physical systems in the field of nonlinear sciences. Full article

In this article, we present the extended simple equations method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The expansions made to the original SEsM algorithm are implemented in several directions: (1) In constructing analytical solutions: exact solutions to FNPDE systems are presented by simple or complex composite functions, including combinations of solutions to two or more different simple equations with distinct independent variables (corresponding to different wave velocities); (2) in selecting appropriate fractional derivatives and appropriate wave transformations: the choice of the type of fractional derivatives for each system of FNPDEs depends on the physical nature of the modeled real process. Based on this choice, the range of applicable wave transformations that are used to reduce FNPDEs to nonlinear ODEs has been expanded. It includes not only various forms of fractional traveling wave transformations but also standard traveling wave transformations. Based on these methodological enhancements, a generalized SEsM algorithm has been developed to derive exact solutions of systems of FNPDEs. This algorithm provides multiple options at each step, enabling the user to select the most appropriate variant depending on the expected wave dynamics in the modeled physical context. Two specific variants of the generalized SEsM algorithm have been applied to obtain exact solutions to two time-fractional shallow-water-like systems. For generating these exact solutions, it is assumed that each system variable in the studied models exhibits multi-wave behavior, which is expressed as a superposition of two waves propagating at different velocities. As a result, numerous novel multi-wave solutions are derived, involving combinations of hyperbolic-like, elliptic-like, and trigonometric-like functions. The obtained analytical solutions can provide valuable qualitative insights into complex wave dynamics in generalized spatio-temporal dynamical systems, with relevance to areas such as ocean current modeling, multiphase fluid dynamics and geophysical fluid modeling. Full article

The water hammer phenomenon, characterized by transient pressure surges due to rapid fluid deceleration in pipelines, poses significant risks to hydraulic systems. Valve closure time is a critical parameter influencing pressure magnitude, necessitating precise calibration to ensure system safety. While numerical methods like the MacCormack scheme provide accurate solutions, their computational intensity limits practical applications. This study addresses this limitation by proposing a machine learning (ML) framework employing a multilayer perceptron (MLP) artificial neural network (ANN) to predict optimal pressure values—defined as the lowest maximum pressure obtained for several closure laws at a given closure time—corresponding to specific valve closure times. The ANN was trained on 637 simulations generated via the MacCormack method, which solves the hyperbolic partial differential equations governing transient flow in a reservoir-pipeline-valve (RPV) system. Performance evaluation metrics demonstrate the ANN’s exceptional robustness and accuracy, achieving a root mean square error (RMSE) of 0.068, Nash-Sutcliffe efficiency (NSE) of 0.99, and a correlation coefficient (R) of 0.99, with a maximum relative error below 1%. The results highlight the ANN’s superior predictive accuracy and flexibility in capturing complex transient flow dynamics, outperforming conventional numerical methods. Notably, the ANN reduced computational time from days for iterative simulations to mere seconds, enabling rapid prediction of pressure-time curves critical for real-time decision-making. This framework offers a computationally efficient and reliable alternative for optimizing valve closure strategies, mitigating water hammer risks, and enhancing pipeline safety. By bridging numerical rigor with machine learning, this work enhances hydraulic infrastructure resilience across industrial and urban networks. Full article

Waves on graphs are a current subject of research interest. As opposed to flows on graphs, the reflection–transmission of waves at the graph’s vertex is a problem that needs to be further modelled mathematically. The literature on the reflection and transmission of waves at a vertex is scarce. Some articles are reviewed and discussed. Water waves are a good topic for comparing different mathematical models, from hyperbolic conservation laws to weakly nonlinear, weakly dispersive systems of partial differential equations on a two-dimensional fattened (thick) graph and the respective one-dimensional graph-model reduction. In this study, we present a particular water wave model in which junction angles are systematically included in the mathematical model. Comparing the solutions with the fattened-graph model gave rise to a more general compatibility condition at the vertex. Current research topics of interest are outlined at the end. Full article

In this study, we propose a generalized framework based on the Simple Equations Method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The key developments over the original SEsM in the proposed analytical framework include the following: (1) an extension of the original SEsM by constructing the solutions of the studied FNPDEs as complex composite functions which combine two single composite functions, comprising the power series of the solutions of two simple equations or two special functions with different independent variables (different wave coordinates); (2) an extension of the scope of fractional wave transformations used to reduce the studied FNPDEs to different types of ODEs, depending on the physical nature of the studied FNPDEs and the type of selected simple equations. One variant of the proposed generalized SEsM is applied to a mathematical generalization inspired by the classical Boussinesq model. The studied time-fractional Boussinesq-like system describes more intricate or multiphase environments, where classical assumptions (such as constant wave speed and energy conservation) are no longer applicable. Based on the applied SEsM variant, we assume that each system variable in the studied model supports multi-wave dynamics, which involves combined propagation of two distinct waves traveling at different wave speeds. As a result, numerous new multi-wave solutions including combinations of different hyperbolic, elliptic, and trigonometric functions are derived. To visualize the wave dynamics and validate the theoretical results, some of the obtained analytical solutions are numerically simulated. The new analytical solutions obtained in this study can contribute to the prediction and control of more specific physical processes, including diffusion in porous media, nanofluid dynamics, ocean current modeling, multiphase fluid dynamics, as well as several geophysical phenomena. Full article

This study presents Galerkin and collocation algorithms based on Telephone polynomials (TelPs) for effectively solving high-order linear and non-linear ordinary differential equations (ODEs) and ODE systems, including those with homogeneous and nonhomogeneous initial conditions (ICs). The suggested approach also handles partial differential equations (PDEs), emphasizing hyperbolic PDEs. The primary contribution is to use suitable combinations of the TelPs, which significantly streamlines the numerical implementation. A comprehensive study has been conducted on the convergence of the utilized telephone expansions. Compared to the current spectral approaches, the proposed algorithms exhibit greater accuracy and convergence, as demonstrated by several illustrative examples that prove their applicability and efficiency. Full article

Miscible gas flooding improves oil displacement through mass exchange between oil and gas phases. It is one of the most efficient enhanced oil recovery methods for intermediate density oil reservoirs. In this work, analytical solutions for saturation, concentration and pressure are derived for oil displacement by a partially miscible gas injection at a constant rate. The mathematical model considers two-phase, three-component fluid flow in a one-dimensional homogeneous reservoir initially saturated by a single oil phase. Phase saturations and component concentrations are described by a $2\times 2$ hyperbolic system of partial differential equations, which is solved by the method of characteristics. Once this Goursat–Riemann problem is solved, the pressure drop between two points in the porous media is obtained by the integration of Darcy’s law. The solution of this problem may present three different fluid regions depending on the rock–fluid parameters: a single-phase gas region near the injection point, followed by a two-phase region where mass transfer takes place and a single-phase oil region. We considered the single-phase gas and the two-phase gas/oil regions as incompressible, while the single-phase oil region may be incompressible or slightly compressible. The solutions derived in this work are applied for a specific set of rock and fluid properties. For this data set, the two-phase region displays rarefaction waves, shock waves and constant states. The pressure behavior depends on the physical model (incompressible, compressible and finite or infinite porous media). In all cases, the injection pressure is the result of the sum of two terms: one represents the effect of the mobility contrast between phases and the other represents the single-phase oil solution. The solutions obtained in this work are compared to an equivalent immiscible solution, which shows that the miscible displacement is more efficient. Full article

In this study, we discuss the dynamical behaviors and extract new interesting wave soliton solutions of the two significant well-known nonlinear partial differential equations (NPDEs), namely, the Korteweg–de Vries equation (KdVE) and the Jaulent–Miodek hierarchy equation (JMHE). This investigation has applications in pattern recognition, fluid dynamics, neural networks, mechanical systems, ecological systems, control theory, economic systems, bifurcation analysis, and chaotic phenomena. In addition, bifurcation analysis and the chaotic behavior of the KdVE and JMHE are the main issues of the present research. As a result, in this study, we obtain very effective advanced exact traveling wave solutions with the aid of the proposed mathematical method, and the solutions involve rational functions, hyperbolic functions, and trigonometric functions that play a vital role in illustrating and developing the models involving the KdVE and the JMHE. These new exact wave solutions lead to utilizing real problems and give an advanced explanation of our mentioned mathematical models that we did not yet have. Some of the attained solutions of the two equations are graphically displayed with 3D, 2D, and contour panels of different shapes, like periodic, singular periodic, kink, anti-kink, bell, anti-bell, soliton, and singular soliton wave solutions. The solutions obtained in this study of our considered equations can lead to the acceptance of our proposed method, effectively utilized to investigate the solutions for the mathematical models of various important complex problems in natural science and engineering. Full article

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order $O(\Delta t+{\overline{h}}^4)$, where $\Delta t$ is the time step and $\overline{h}$ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method. Full article

The particle size distribution (PSD) in emulsion polymerization (EP) has been modeled in the past using either the pseudo bulk (PB) or the 0-1/0-1-2 approaches. There is some controversy on the proper type of model to be used to simulate the experimental PSDs, which are apparently broader than the theoretical ones. Additionally, the numerical technique employed to solve the model equations, involving hyperbolic partial differential equations (PDEs) with moving and possibly steep fronts, has to be precise and robust, which is not a trivial matter. A deterministic kinetic model for the PSD evolution of ab initio EP of vinyl monomers was developed to investigate these issues. The model considers three phases, micellar nucleation, and particles that can contain $n\ge 0$ radicals. Finite volume (FV) and weighted-residual methods are used to solve the system of PDEs and compared; their limitations are also identified. The model was validated by comparing predictions with data of monomer conversion and PSD for the batch emulsion homopolymerization of styrene (Sty) and methyl methacrylate (MMA) using sodium dodecyl sulfate (SDS)/potassium persulfate (KPS) at 60 °C, as well as the copolymerization of Sty-MMA (50/50; mol/mol) at 50 and 60 °C. It is concluded that the PB model has a structural problem when attempting to adequately represent PSDs with steep fronts, so its use is discouraged. On the other hand, there is no generalized evidence of the need to add a stochastic term to enhance the PSD prediction of EP deterministic models. Full article

The telegraph equation is a hyperbolic partial differential equation that has many applications in symmetric and asymmetric problems. In this paper, the solution of the time-fractional telegraph equation is obtained using a hybrid method. The numerical simulation is performed based on a combination of the finite difference and differential transform methods, such that at first, the equation is semi-discretized along the spatial ordinate, and then the resulting system of ordinary differential equations is solved using the fractional differential transform method. This hybrid technique is tested for some prominent linear and nonlinear examples. It is very simple and has a very small computation time; also, the obtained results demonstrate that the exact solutions are exactly symmetric with approximate solutions. The results of our scheme are compared with the two-dimensional differential transform method. The numerical results show that the proposed method is more accurate and effective than the two-dimensional fractional differential transform technique. Also, the implementation process of this method is very simple, so its computer programming is very fast. Full article

In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs). Full article

We extend our analysis on the Lie symmetries in fluid dynamics to the case of macroscopic traffic estimation models. In particular we study the Aw–Rascle–Zhang model for traffic estimation, which consists of two hyperbolic first-order partial differential equations. The Lie symmetries, the one-dimensional optimal system and the corresponding Lie invariants are determined. Specifically, we find that the admitted Lie symmetries form the four-dimensional Lie algebra $A_{4,12}$. The resulting one-dimensional optimal system is consisted by seven one-dimensional Lie algebras. Finally, we apply the Lie symmetries in order to define similarity transformations and derive new analytic solutions for the traffic model. The qualitative behaviour of the solutions is discussed. Full article

In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the $\chi$-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively. Full article

We propose a differential quadrature method (DQM) based on cubic hyperbolic B-spline basis functions for computing 3D wave equations. This method converts the problem into a system of ODEs. We use an optimum five-stage and order four SSP Runge-Kutta (SSPRK-(5,4)) scheme to solve the obtained system of ODEs. The matrix stability analysis is also investigated. The accuracy and efficiency of the proposed method are demonstrated via three numerical examples. It has been found that the proposed method gives more accurate results than the existing methods. The main purpose of this work is to present an accurate, economically easy-to-implement, and stable technique for solving hyperbolic partial differential equations. Full article