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This article represented the investigation of the modified mixed Korteweg-de Vries equation using different versatile approaches. First, the Lie point symmetry approach was used to determine all possible symmetry generators. With the help of these generators, we reduced the dimension of the proposed equation which leads to the ordinary differential equation. Second, we employed the unified Riccati equation expansion technique to construct the abundance of soliton dynamics. A group of kink solitons and other solitons related to hyperbolic functions were among these solutions. To give the physical meaning of these theoretical results, we plotted these solutions in 3D, contour, and 2D graphs using suitable physical parameters. The comprehend outcomes were reported, which can be useful and beneficial in the future investigation of the studied equation. The results showed that applied techniques are very useful to study the other nonlinear physical problems in nonlinear sciences.

Abstract To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

This paper aims to explore and examine a fractional differential equation in the fuzzy conformable derivative sense. To achieve this goal, a novel analytical algorithm is formulated based on the Laplace-residual power series method to solve the fuzzy conformable fractional differential equations. The methodology being used to discover the fuzzy solutions depends on converting the desired equations into two fractional crisp systems expressed in $ \wp $-cut form. The main objective of our algorithm is to transform the systems into fuzzy conformable Laplace space. The transformation simplifies the system by reducing its order and turning it into an easy-to-solve algorithmic equation. The solutions of three important applications are provided in a fuzzy convergent conformable fractional series. Both the theoretical and numerical implications of the fuzzy conformable concept are explored about the consequential outcomes. The convergence analysis and theorems of the developed algorithm are also studied and analyzed in this regard. Additionally, this article showcases a selection of results through the use of both two-dimensional and three-dimensional graphs. Ultimately, the findings of this study underscore the efficacy, speed, and ease of the Laplace-residual power series algorithm in finding solutions for uncertain models that arise in various physical phenomena.

An integro‐differential equation for the probability density of the generalized stochastic Ornstein–Uhlenbeck process with jump diffusion is considered for a special case of the Laplacian distribution of jumps. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly, as a finite sum. Alternatively, the fundamental solution can be represented as converging power series. The properties of this solution are investigated. The fundamental solution makes it possible to obtain explicit formulas for the density at each instant of time, which is important, for example, for testing numerical methods. [ABSTRACT FROM AUTHOR]

The current manuscript proposes an innovative approach for obtaining exact solutions to the conformable time-fractional nonlinear differential-difference equations (NDDEs). The fundamental concept of this approach involves the generalized exponential rational function method (GERFM). In this method, the exact solutions include trigonometric and hyperbolic functions. In order to assess this method's efficacy, we consider its application to the conformable time-fractional Hybrid Lattice equation. After solving this model, we provide its dynamic behavior through the 3D, 2D, and Contour graphs. The soliton solutions obtained are used to describe any related physical phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exact solutions of a modified generalized multidimensional fractional Kadomtsev–Petviashvili (KP) equation by the bifurcation method. First, the equation is converted into a planar dynamical system through fractional complex wave transformation. The phase portraits of the equation and qualitative analysis are presented under different bifurcation conditions. Then, the bounded and unbounded traveling wave solutions, including periodic, kink, anti-kink, dark-solitary, bright-solitary and breaking wave solutions, are acquired by integrating along different orbits. Finally, numerical simulations of the dynamic behaviors of the solutions obtained are graphically illustrated by choosing appropriate parameters. [ABSTRACT FROM AUTHOR]

In this paper, the possibility of existing a novel class of compact charged spheres based on a charged perfect fluid within the realm of f(Q)$f(Q)$ gravity theory is explored. The authors started by proposing physically meaningful explicit formulas for the potential, denoted λ(r)$\lambda (r)$, and the electric field to find a close‐form solution. More precisely, the change of the dependent variable approach by exploiting the transformation Y(r)=eζ(r)/2$Y(r)=e^{\zeta (r)/2}$ is applied. Successively, the field equations analytically are solved and generate the most general solution, which leads us to examine various significant aspects of the stellar system. These aspects comprise the regularity of gravitational potentials, energy density and pressure, electric charge, the mass‐radius relationship, subluminal sound velocities in the radial direction, and the adiabatic index for charged compact stars. For a more in‐depth system study, mass measurements using contour diagrams are carried out. This mainly involves varying the variable parameters β1$\beta _1$ and E0$E_0$ to distinguish their effect on the mass distribution within the stellar structure. What is more, the electric charge controls the stability of the stellar system is shown, which yields that a stable system can possess a maximum charge of order 1020C$10^{20}\nobreakspace \text{C}$. The results strongly argue that charged stars could conceivably exist in nature and that such a deviation from traditional theories may be seen in future astrophysical observations. In this paper, the possibility of existing a novel class of compact charged spheres based on a charged perfect fluid within the realm of f(Q)$f\!(\mathcal{Q})$ gravity theory is explored. The authors started by proposing physically meaningful explicit formulas for the potential, denoted λ$\lambda$(r), and the electric field to find a close‐form solution. More precisely, the change of the dependent variable approach by exploiting the transformation Y(r) = eζ(r)∕2 is applied. Successively, the field equations analytically are solved and generate the most general solution, which leads us to examine various significant aspects of the stellar system. These aspects comprise the regularity of gravitational potentials, energy density and pressure, electric charge, the mass‐radius relationship, subluminal sound velocities in the radial direction, and the adiabatic index for charged compact stars. For a more in‐depth system study, mass measurements using contour diagrams are carried out. This mainly involves varying the variable parameters β1 and E0 to distinguish their effect on the mass distribution within the stellar structure. What is more, the electric charge controls the stability of the stellar system is shown, which yields that a stable system can possess a maximum charge of order 1020 C. The results strongly argue that charged stars could conceivably exist in nature and that such a deviation from traditional theories may be seen in future astrophysical observations. [ABSTRACT FROM AUTHOR]

Dynamics of cracked nanobeams surrounded by size-dependent Winkler –Pasternak medium are investigated utilizing the two-phase local/nonlocal elasticity, as a paradox-free form of the nonlocal theory, in order to considering the nonlocal effect on both of the defected nanobeam and the two-parameter elastic medium. To this aim, governing equations as well as all boundary and compatibility conditions including the constitutive ones corresponding to the two-phase cracked nanobeam and the two-phase medium are derived. The exact solution is presented by combining the governing equations and satisfying all higher order boundary conditions. To confirm the validity of the present formulation and results, several comparison studies are established in detail. The influences of different parameters such as local phase fraction factor, crack characteristics and nonlocal parameter on vibration frequencies are discussed. The present results reveal that applying size dependency on the Winkler –Pasternak medium surrounding the nanobeams leads to significant changes in the vibration frequencies of both intact and defected nanobeams. Also, impact of nonlocal effects can increase or decrease in the defected nanobeams depending on the crack characteristics. This research can be useful to achieve more accurate predictions in vibration analysis of defected nanostructures embedded in two-parameter medium. [ABSTRACT FROM AUTHOR]

Using the bilinear neural network method (BNNM) and the symbolic computation system Mathematica, this paper explains how to find an exact solution for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. In terms of activation function and weight coefficient, BNNM is a more appealing option for users than traditional symbolic computation methods. It is possible to develop a wide range of solutions and expand the classes of exact solutions by modifying the activation function. The activation function’s versatility allows it to generate a wide range of solutions with several theoretical and practical uses. The analytical solution is obtained by using a double layer type, while the rogue wave solution and mixed solutions are obtained by using a single layer type. The evolution of these waves is then illustrated using various 3D graphs, 2D graphs, and density plots. [ABSTRACT FROM AUTHOR]

The steady pipe flow of thixotropic yield stress fluids has been investigated theoretically based on a modified isotropic kinematic hardening (mIKH) model. Analytical solution is derived for a specific case (m = n = 1) and a general semi-analytical solution is put forward as well. The effect of thixotropic yield stress on shear rate and velocity profiles is illustrated by comparing to other well-known solutions. Moreover, the influences of model parameters are examined. It is worth noting that shear banding may occur at the yielded surface in case of a sufficiently large Bingham number, thixotropic number, and flow index, but a sufficient small value of structure-related exponent. [ABSTRACT FROM AUTHOR]

The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert method. Compared to the local HOGI equation, the solutions of nonlocal equations not only depend on the local spatial and time variables, but also the nonlocal variables. To exhibit the dynamic behavior, we consider the reverse-spacetime multi-component HOGI equation and its Riemann–Hilbert problem. When the Riemann–Hilbert problem is regular, the integral form solution can be given. Conversely, the exact solutions can be obtained explicitly. Finally, as concrete examples, the periodic solutions of the two-component nonlocal HOGI equation are given, which is different from the local equation. [ABSTRACT FROM AUTHOR]

The homotopy perturbation method (HPM) is one of the recent fundamental methods for solving differential equations. However, checking the accuracy of this method has been ignored by some authors in the literature. This paper reanalyzes the nonlinear system of ordinary differential equations (ODEs) describing the SIR epidemic model, which has been solved in the literature utilizing the HPM. The main objective of this work is to obtain a highly accurate analytical solution for this model via a direct technique. The proposed technique is mainly based on reducing the given system to a single nonlinear ODE that can be easily solved. Numerical results are conducted to compare our approach with the previous HPM, where the Runge–Kutta numerical method is chosen as a reference solution. The obtained results reveal that the current technique exhibits better accuracy over HPM in the literature. Moreover, some physical properties are introduced and discussed in detail regarding the influence of the transmission rate on the behavior of the SIR model. [ABSTRACT FROM AUTHOR]

An interface crack between dissimilar one-dimensional hexagonal quasicrystals with piezoelectric effect under anti-plane shear and in-plane electric loadings is considered. Mixed boundary conditions at the crack faces are studied. Using special representations of field variables via sectionally analytic vector-functions, a homogeneous combined Dirichlet–Riemann boundary value problem and a Hilbert problem are formulated. Exact analytical solutions of both these problems are obtained, and analytical expressions for the phonon and phason stresses and the electric field as well as for the derivative jumps of the phonon and phason displacements and also the electrical displacement jump along the bimaterial interface are derived. The field intensity factors are determined as well. The dependencies of the mentioned values on the magnitude and direction of the external electric loading and different ratios of electrically conductive and electrically permeable crack face zone lengths are demonstrated in graph and table forms. [ABSTRACT FROM AUTHOR]

An analytical solution is presented for the axisymmetric static responses of thick functionally graded (FGM) cylindrical shells with general boundary conditions, which strictly satisfies the boundary conditions along the thickness. The illustrative examples show that the presented solution of thick cylindrical shells under various boundary conditions is of high precision and good convergence. The exact distributions of displacement and stress components along the radial and axial directions are revealed, and the influences of thickness-to-radius ratio, span-to-radius ratio and material gradient index on the static responses of the thick FGM cylindrical shell are analyzed. [ABSTRACT FROM AUTHOR]

An exact solution of an initial value problem for the Susceptible–Exposed–Infectious–Recovered–Deceased (SEIRD) epidemic model is derived, and various properties of the exact solution are obtained. It is shown that the parametric form of the exact solution satisfies some linear differential system including a positive solution of an Abel differential equation of the second kind. In this paper Abel differential equations play an important role in establishing the exact solution of the SEIRD differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an initial value problem for an Abel differential equation. Uniqueness of positive solutions of an initial value problem to SEIRD differential system is also investigated, and it is shown that the exact solution is a unique solution in the class of positive solutions.

In this paper, new results were presented on the symmetry reduction of gas dynamics system of partial differential equations following the general framework of Lev Ovsyannikov's article "The "podmodeli" program, Gas dynamics." We considered the gas dynamics equations with an equation of state prescribing the pressure as the sum of density function and entropy function. This system has a 12-dimensional Lie algebra and we considered its certain three-dimensional subalgebra generated by space translations, Galilean translations and pressure translation. For this subalgebra, the symmetry reduction of the original system leads to a system of ordinary differential equations. We obtained a family of exact solutions for this system, which describes the motion of particles with a linear velocity field and non-homogeneous deformation in the 3D-space. For these solutions, the trajectories of all points are either parabolas or rays. At $ t = 0 $ an instantaneous collapse occurs when all of the particles accumulate in a plane with infinitely many particles at every point of the plane. For a fixed period of time, the particles were emitted from the same point on a plane and ended up on the same line. The gas motion was vortex. A one-dimensional subalgebra embedded into three-dimensional subalgebra was considered. The invariants were written in a consistent form. It was shown that the submodel of rank one was embedded in the submodel of rank three.

Nonlinear evolution equations are employed in the representation of diverse intricate physical events, and the identification of precise solutions for these equations holds significance about their practical implementations. One of the significant challenges is the identification of traveling wave solutions inside established nonlinear evolution systems in the field of mathematical physics. In the present research, we employ the modified sub-equation approach, a very effective and strong technique, to ensure the solutions for the Klein–Gordon featuring cubic nonlinearity and Zakharov Kuznetsov–Benjamin Bona Mahony equations. Several restriction requirements that ensure the existence of these solutions are emphasized. By employing a linearization technique, we ascertain the stability gain. This methodology acquires original precise solutions of soliton nature. Furthermore, the nonlinear wave structures of both equations are illustrated through the consideration of several three-dimensional and two-dimensional plots. These plots are generated by selecting appropriate values for the parameters. It is expected that these innovative solutions would facilitate an in-depth understanding of the evolution and fluidity of these models. The solutions obtained comprise periodic functions, mixed periodic functions, rational solutions, and exponential solutions.

This theoretical analysis provided the exact solution of a steady flow of Casson rheological fluid in fluid-particle suspension models through a divergent channel with consideration of porous medium, electric, and magnetic fields, and slip boundary conditions. The thermal transport analysis is also observed with the consideration of viscous dissipation and uniform heat source. The suitable transformation is used to reduce the partial differential equation into ordinary differential equations and obtain the exact solution by adopting the mathematical software MATHEMATICA 12.0. The momentum and thermal profiles are decreasing functions of the magnetic field parameter. The number of streamlines is increased and covers more parts of the channel for increasing the Darcy force and velocity slip parameters. The computational results of this study will help to understand the momentum and thermal analysis in the fluid-particle suspension model. The results of the current study are useful to increase the oil recovery system, in thermal transport energy, energy production, cooling and heating systems, etc. The current model can be useful in renewable energy to store thermal energy by using the hafnium nanoparticles. The present analysis is original and has not been submitted nor published before. [ABSTRACT FROM AUTHOR]

In this paper, we analyze how Eyring fluid film behavior on a vertically upward moving slippery flat plate can help predictive models in engineering, notably in coating and lubrication operations. This paper deals with the stagnant points and uniform film analysis of Eyring fluid film flow on a vertically upward moving slippery flat plate. The formulated ordinary differential equation is solved for exact analytic solution. Exact analytic expressions for velocity, flow rate, average velocity, shear stress components, and stagnant points are derived. A highly nonlinear algebraic equation is derived for film thickness. Equation is solved by Newton’s method using Maple code. The thickness of film widens with increasing relaxation time, constant plate velocity, and fluid density, while it decreases with increasing fluid viscosity and constant slip parameter. The analysis delineates that the positions of stagnant points tend to relocate closer the slippery plate as the Stokes number, Deborah number, and constant slip parameter increase. A high Deborah number enhances drainage, causing stagnant points to relocate closer to the slippery plate and the development of a stable elastic layer adjacent to the plate. For stagnant points and fluid film thickness, a comparison between the Eyring fluid and existing studies (EPTT, LPTT, UCM, and Newtonian fluid) is also provided. The validity of this paper with the existing studies is also presented by reducing Eyring fluid model to Newtonian model. The results of this research are significant for a wide range of biofluid applications, including agrochemical uses, paint and surface coating flow behavior, thin films on the cornea and lungs, and chemical and nuclear reactor design. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a degenerating parabolic heat equation with nonlinearities of a general type in the presence of a source (sink) in the case of two spatial variables. The problem of initiating a heat wave propagating over a cold (zero) background with a finite velocity and a boundary condition specified on a moving manifold—a closed line—is considered. For this problem, a new existence and uniqueness theorem is proved, a numerical algorithm for constructing a solution based on the boundary element method, collocation method, and difference time approximation is proposed; a special change of variables of the hodograph-type transformation is used. New exact solutions to this equation in the case of power nonlinearities are found. A numerical algorithm is implemented, and a numerical experiment is carried out. A comparison of the constructed numerical solutions with exact ones (found both in this paper and earlier) showed good agreement. The numerical convergence in the time step and number of collocation points is proved. [ABSTRACT FROM AUTHOR]

Phenomena in physics, plasma physics, optical fibers, chemical physics, fluid mechanics, and many fields are often described by the nonlinear evolution equations. The analytical solutions of these equations are very important to understand the evaluation of the physical models. In this paper, the Boiti-Leon-Manna-Pempinelli (BLMP) nonlinear partial differential equation, which can be used to describe the incompressible fluid flow, is analytically studied by using the five different techniques which are direct integration, (G' / G)-expansion method, different form of the (G' / G)-expansion method, two variable (G' / G, 1 / G)-expansion method, and (1 / G')- expansion method. Hyperbolic, trigonometric and rotational forms of solutions are obtained. Our solutions are reduced to the well-known solutions found in the literature by assigning the some special values to the constants appeared in the analytic solutions. Moreover, we have also obtained the new analytic solutions of the BLMP equation. [ABSTRACT FROM AUTHOR]

The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into original analytical solutions of the (2+1)-dimensional combined potential Kadomtsev–Petviashvili and B-type Kadomtsev–Petviashvili equations. For this purpose, the generalized Kudryashov technique (GKT) and exponential rational function technique (ERFT) have been applied to deal with the equation. These two methods have been applied to the model for the first time, and the the generalized Kudryashov method has an important place in the literature. The characteristics of solitons are unveiled through the use of three-dimensional, two-dimensional, contour, and density plots. Furthermore, we conducted a stability analysis on the acquired results. The results obtained in the article were seen to be different compared to other results in the literature and have not been published anywhere before. [ABSTRACT FROM AUTHOR]

Within the framework of the Oberbeck–Boussinesq approximation, the exact solution of equations of thermoconcentration convection is studied, which has the group origin. The issue of applicability of the exact solution for describing steady‐state convective flows of a liquid and a co‐current gas flux under the conditions of inhomogeneous evaporation of the diffusive type in a flat horizontal channel is discussed. Algorithms for obtaining analytical representations of the required functions for various types of the boundary conditions for the temperature function on the outer channel wall are proposed. The influence of the external thermal load and boundary thermal conditions on the structure of the velocity and temperature fields, evaporation mass flow rate, and vapor content in the gas layer is investigated at an example of the HFE‐7100–nitrogen system. The solution correctly predicts hydrodynamical, temperature, and concentration parameters of convective regimes appearing in the two‐phase system. The main characteristics are compared with the characteristics of the system in the case of uniform evaporation with a constant intensity. [ABSTRACT FROM AUTHOR]

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called 'diffusion-wave-type solutions'. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the Riemann problem for two‐layered blood flow model, which is represented by a system of quasi‐linear hyperbolic partial differential equations (PDEs) derived from the Euler equations by vertical averaging across each layer. We consider the Riemann problem with varying velocities and equal constant density through arteries. For instance, the flow layer close to the wall of vessel has a slower average speed than the layer far from the vessel because of the viscous effect of the blood vessel. We first establish the existence and uniqueness of the corresponding Riemann solution by a thorough investigation of the properties of elementary waves, namely, shock wave, rarefaction wave, and contact discontinuity wave. Further, we extensively analyze the elementary wave interaction between rarefaction wave and shock wave with contact discontinuity and rarefaction wave and shock wave. The global structure of the Riemann solutions after each wave interaction is explicitly constructed and graphically illustrated towards the end. [ABSTRACT FROM AUTHOR]

Population balance equations may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. In this paper, a (2 + 1) dimensional population balance equation with aggregation, nucleation, growth and breakage processes is solved analytically by use of the methods of scaling transformation group, observation and trial function. Symmetries, reduced equations, invariant solutions, exact solutions, existence of solutions, evolution analysis of dynamic behavior for solutions are presented. The exact solutions obtained can be compared with the numerical scheme. The obtained results also show that the method of scaling transformation group can be applied to study integro-partial differential equations. [ABSTRACT FROM AUTHOR]

The standard pantograph delay equation (SPDDE) is one of the famous delay models. This standard model is basically homogeneous in nature and it has been extensively studied in the literature. However, the studies on the general inhomogeneous form of such a model seem rare. This paper considers the inhomogeneous pantograph delay equation (IPDDE) with a kind of arbitrary inhomogeneous term. This arbitrary inhomogeneous term is used in different forms to generate various classes of IPDDEs. The solutions of the present classes are obtained in closed series forms which satisfy the criteria of convergence. Also, the exact solutions are determined for these classes under a certain relation between the given initial condition of the model and the initial value of the inhomogeneous term. Several classes are generated and solved when the inhomogeneous term takes the form of trigonometric, exponential, and hyperbolic functions. Some existing results in the literature are recovered as special cases of the present ones. Moreover, the behaviors of the obtained solutions are demonstrated through graphs for various kinds of IPDDEs. [ABSTRACT FROM AUTHOR]

Non-reciprocal interactions of discrete or continuous systems may induce surprising responses such as flutter instabilities. It is shown in this paper that a finite one-dimensional lattice under non-symmetrical elastic interactions may flutter for sufficiently strong unsymmetrical interactions. An exact solution is presented for the vibration of such one-dimensional lattices with direct and non-symmetrical elastic interactions. An internal force controlling the interactions is included in the model as an additional force for each mass, which acts proportionally to the elongation of a spring at its position. This non-conservative problem due to this circulatory interaction is solved from the resolution of a linear difference equation for this unsymmetrical repetitive lattice. It is possible to derive the exact eigenfrequency dependence with respect to the unsymmetrical interaction parameter, which plays the role of a bifurcation parameter. Divergence and flutter instabilities of this fixed–fixed non-conservative axial lattice under non-Hermitian interactions are theoretically predicted, from a direct approach or by solving the difference equation whatever the number of masses of the lattice. It is shown that the system may flutter for sufficiently strong unsymmetrical interactions, whatever the size of the system, for even or odd number of masses. However, divergence instability may arise in such a system only for even number of masses. The drastic change of response of the present system for odd or even number of particles is specific of the discrete nature of the dynamic system. [ABSTRACT FROM AUTHOR]

In this work, the perturbed nonlinear Schrödinger's equation (NLSE) with self-phase modulation (SPM) of Kudryashov's sextic power law which describes the soliton's propagation in optical fiber is investigated. New exact solutions will be introduced including doubly periodic solutions in the form of Weirstrass and hyperbolic functions by using the simple equation method. There have been several different kinds of optical soliton solutions proposed, including dark, bright, kink, and anti-kink soliton solutions. To further understand some of its physical characteristics, the resulting optical soliton solutions are illustrated graphically. [ABSTRACT FROM AUTHOR]

Carbon nanotubes (CNTs) nanofluids are gaining increased popularity among researchers due to their outstanding thermal properties, leading to numerous promising industrial applications. Analytical solutions discovered in the study of CNTs nanofluids, combined with a Casson-type fluid model, are extremely limited. Therefore, a study on the heat transfer analysis of an unsteady and incompressible Casson carbon nanofluid flow is conducted. Human blood-based single-walled carbon nanotubes (SWCNTs) and human blood-based multi-walled carbon nanotubes (MWCNTs) are considered as nanofluids that move beyond an exponentially accelerated vertical plate. A set of dimensional momentum and energy equations, along with their initial and exponentially accelerated boundary conditions, is employed to represent the problem. The transformation of these equations to the dimensionless expression is achieved by using suitable dimensionless variables. The resulting equations are then tackled using Laplace transformation to acquire the analytical solution for temperature and velocity. Figures and tables are produced for a further analysis of temperature and velocity characteristics. The study shows that an increase in nanoparticle volume fraction enhances nanofluid flow and heat transmission, proving highly beneficial for cancer treatment. However, the flow is retarded due to the increment of Casson parameter values, while an enhancement is observed with a superior accelerating parameter. [ABSTRACT FROM AUTHOR]

Our research focuses on a fourth-order partial differential equation (PDE) that arises from the Timoshenko model for beams. This PDE pertains to situations where the elastic moduli remain constant and an external load, represented as F, is applied. We thoroughly analyze Lie symmetries and categorize the various types of applied forces. Initially, the principal Lie algebra is two-dimensional, but in certain noteworthy cases, it extends to three dimensions or even more. For each specific case, we derive the optimal system, which serves as a foundation for symmetry reductions, transforming the original PDE into ordinary differential equations. In certain instances, we successfully identify exact solutions using this reduction process. Additionally, we delve into the conservation laws using a direct method proposed by Anco, with a particular focus on specific classes within the equation. The findings we have presented in our study are indeed original and innovative. This study serves as compelling evidence for the robustness and efficacy of the Lie symmetry method, showcasing its ability to provide valuable insights and solutions in the realm of mathematical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider how to find new exact solutions for nonlinear partial differential equations using the extended Kudryashov method. This method mainly uses the Riccati equation and the Bernoulli equation where there are some underdetermined constant parameters. And we also use the concept of symmetry to study its reduction equation, Lie transformation group, self-adjointness, and conservation laws. This paper mainly studies the Boussinesq class and the shallow water wave equation in (1 + 1) dimensions and tries to find new exact solutions and symmetry properties of them. [ABSTRACT FROM AUTHOR]

This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional derivative of first type considers the lower bound as a zero while the second type applies negative infinity as a lower bound. Due to the differences in properties of the two operators, two different solutions are obtained for the present two classes of fractional differential equations under appropriate initial conditions. It is shown that the zeroth lower bound implies implicit solutions in terms of the Mittag–Leffler functions while explicit solutions are derived when negative infinity is taken as a lower bound. Such explicit solutions are obtained for the current two classes in terms of trigonometric and hyperbolic functions. Some theoretical results are introduced to facilitate the solutions procedures. Moreover, the characteristics of the obtained solutions are discussed and interpreted. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define the generalized discrete proportional derivative (GDPD) and illustrate the application of the Leibniz theorem, the binomial expansion, and Montmort's formulas in the context of the generalized discrete proportional case. Furthermore, we introduce the generalized discrete proportional Laplace transform and determine the GDPLT of various functions using the inverse operator. The results obtained are showcased through relevant examples and validated using MATLAB. [ABSTRACT FROM AUTHOR]

In this work, we extend the analysis of the relativistic Dirac–Rosen–Morse problem in curved space–time. For that, we consider the Dirac equation in curved space–time with line element ds2 = (1 + α2U(r))2(dt2 − dr2) − r2dθ2 − r2sin 2θdϕ2, where α is fine structural constant, U(r) is a scalar potential, and in the presence of the electromagnetic field Aμ = (V(r), cA(r), 0, 0). Because of the spherical symmetry, the angular spinor is given in terms of the spherical harmonics. For the radial spinor, we apply a unitary transformation and define the vector component of the electromagnetic field A(r) written as a function of V(r) and U(r) so as to solve the radial spinor for Dirac–Rosen–Morse problem. Graphical analyses were performed comparing the eigenenergies and the probability densities in curved and flat space–time to visualize the influence of curvature in space–time on the two-component radial spinor, with the upper and lower components representing the particle and antiparticle, respectively. [ABSTRACT FROM AUTHOR]

Today, due to the many applications of sandwich beams in industry, studying the forced vibrations of these structures is important. In these structures, amplifiers are used to improve mechanical properties. In this paper, metal-based graphene (copper) is used to improve the mechanical properties. This research uses a functionally graded (FG) graphene-reinforced copper-based composite (GRCC) sandwich beam and FG soft porous core, subjected to two moving loads and located on an elastic foundation. The equations are derived using Soldatos' higher-order shear deformation theory in axial and transverse directions. The exact problem under the magnetic field is solved using the Laplace method, which has not been done. The advantages of this method are the simplicity of solving and reducing to zero the error percentage that exists in numerical solutions. The results are compared with previous works. Finally, the effect of various parameters such as magnetic, porosity coefficient, elastic constant, thickness ratio, and velocity of moving load on the dynamic response of the sandwich beam is investigated. It should be noted that the results can be used to construct this type of structure. [ABSTRACT FROM AUTHOR]

Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either report numerical solutions or require an assumption of linear diffusion to determine exact solutions. Unfortunately, numerical solutions do not reveal the relationship between the model parameters and the solution features. Additionally, experimental observations typically report the presence of sharp fronts, which are not captured by linear diffusion. Here we address both limitations by presenting exact sharp-fronted solutions to a model of degenerate nonlinear diffusion on a growing domain. We obtain the solution by identifying a series of transformations that converts the model of a nonlinear diffusive process on an evolving domain to a nonlinear diffusion equation on a fixed domain, which admits known exact solutions for certain choices of diffusivity functions. We determine expressions for critical time scales and domain growth rates such that the diffusive population never reaches the domain boundaries and hence the solution remains valid. [ABSTRACT FROM AUTHOR]

Equations describing nonstationary and stationary flows of an incompressible polymer fluid through a pipe are derived based on the rheological mesoscopic Pokrovskii–Vinogradov model. Their exact stationary solutions are obtained and conditions providing their existence are outlined. Numerical simulation of the stabilization of a nonstationary flow is carried out and the restrictions on the values of parameters that ensure stabilization are computed. In a number of cases these restrictions coincide with the conditions of the existence of stationary solutions. The obtained results enable us to describe constructively the process of destruction of laminar Poiseuille-type flows, which usually initiates the onset of turbulence. The key role in mechanics of this process is played by the size and orientation of macromolecules of the polymer fluid. The mathematical description of the process uses essentially the solutions' singular points. [ABSTRACT FROM AUTHOR]

This article represented the investigation of the modified mixed Korteweg-de Vries equation using different versatile approaches. First, the Lie point symmetry approach was used to determine all possible symmetry generators. With the help of these generators, we reduced the dimension of the proposed equation which leads to the ordinary differential equation. Second, we employed the unified Riccati equation expansion technique to construct the abundance of soliton dynamics. A group of kink solitons and other solitons related to hyperbolic functions were among these solutions. To give the physical meaning of these theoretical results, we plotted these solutions in 3D, contour, and 2D graphs using suitable physical parameters. The comprehend outcomes were reported, which can be useful and beneficial in the future investigation of the studied equation. The results showed that applied techniques are very useful to study the other nonlinear physical problems in nonlinear sciences. [ABSTRACT FROM AUTHOR]

This paper aims to explore and examine a fractional differential equation in the fuzzy conformable derivative sense. To achieve this goal, a novel analytical algorithm is formulated based on the Laplace-residual power series method to solve the fuzzy conformable fractional differential equations. The methodology being used to discover the fuzzy solutions depends on converting the desired equations into two fractional crisp systems expressed in P-cut form. The main objective of our algorithm is to transform the systems into fuzzy conformable Laplace space. The transformation simplifies the system by reducing its order and turning it into an easy-to-solve algorithmic equation. The solutions of three important applications are provided in a fuzzy convergent conformable fractional series. Both the theoretical and numerical implications of the fuzzy conformable concept are explored about the consequential outcomes. The convergence analysis and theorems of the developed algorithm are also studied and analyzed in this regard. Additionally, this article showcases a selection of results through the use of both two-dimensional and three-dimensional graphs. Ultimately, the findings of this study underscore the efficacy, speed, and ease of the Laplace-residual power series algorithm in finding solutions for uncertain models that arise in various physical phenomena. [ABSTRACT FROM AUTHOR]

Purpose: This is the first investigation to employ an exact solution to explore the free vibrational behavior of beams under the impact of flexoelectricity and temperature when the beam is maintained by a viscoelastic foundation and subjected to axial compression. Methods: This study makes use of the analytical solution. Results: This study provided a solution in the form of an equation for estimating the natural frequency of oscillations of a beam resting on a viscoelastic substrate while accounting for the flexoelectric effect. This is the equation used to verify the accuracy of various approximation methods (FEM, isogeometric approach, etc.), and it is simple to use in reality to estimate the nanobeams' natural frequency quickly. In addition, the study findings provide a foundation for tackling more difficult issues and aid engineers in the design, production, and practical use of nanobeams. Conclusion: The article presented the exact solution to study the specific vibration of the beam, while it was resting on the viscoelastic medium while considering influences of the flexoelectricity. In this solution, effects of temperature and the influence of mechanical compressive force acting along the beam axis are both taken into account. The calculation formulae for beams are taken from the classical beam theory. When viscoelastic foundations are included, the particular vibrational response of the beam will comprise both real and imaginary parts. This discovery is quite fascinating, and it provides a platform for additional research into the mechanical reaction of beams when they are supported by a viscoelastic foundation. [ABSTRACT FROM AUTHOR]

This investigation is an approach to setup an analytical solution of steady plane allied MHD fluid flow having infinite electrical conductivity in a rotating frame through porous media by Martin's method. The governing non-linear equations of the fluid flow are transformed into a new form called Martin's form by employing differential geometry where the curvilinear co-ordinates (Φ, Ψ) in the plane of flow shows that, the co-ordinate lines Ψ are the streamlines of flow and the co-ordinate lines Φ are arbitrary constants. Exact solution is obtained and velocity, vorticity, current density magnetic field and pressure distribution are found out. Also, the diagrams have been plotted to sketch the streamline patterns and to study variation of pressure function with angular velocity. [ABSTRACT FROM AUTHOR]

This paper employs an analytical approach to achieve a precise solution and physical application for the unsteady one-dimensional adiabatic flow of weak shock waves with generalized geometries in a non-viscous perfect fluid under the influence of a weak gravitational field. In the disturbed region, the density is considered to have a functional relationship with distance, meaning that a relative change in distance from the source of disturbance causes a corresponding change in density. Finally, the problem's solution comes in the shape of distance and time power. The current technique handles this scenario in a natural way, and the approximations produce results that are reasonably accurate. [ABSTRACT FROM AUTHOR]

We investigated the (2+1)D nonlinear Rossby waves with variable coefficients. When the group velocity is considered as a function of time in the stretch coordinates of employing multi-scale analysis, and the weak nonlinear perturbation expansions are used, the variable coefficient (2+1)D Kadomtsev-Petviashvili equation describing Rossby waves was derived from the quasi-geostrophic potential vorticity equation. For studying the effect of variable coefficients, Rossby soliton wave is obtained via auxiliary equation method. And from the results in Sect. 3.1, the variable coefficients cause not only the propagation of Rossby wave, but also the increase of its amplitude. For analyzing the rogue wave, we utilize the modified Hirota bilinear method, which has the advantage of structuring one test functions and calculating one time. Interestingly, the variable coefficients are limited to constants in Sect. 3.2. The physics for the evolutions of Rossby waves are analyzed. Through numerical simulations, the results show all blocking structures in this article move in longitude, and the blocking structures caused by soliton, lump and interaction wave, are depicted in the Rossby wave flow field. [ABSTRACT FROM AUTHOR]

A novel method is presented for reproducing kernel of a 2D fractional diffusion equation. The exact solution is expressed as a series, which is then truncated to get an approximate solution. In addition, some techniques to improve existing methods are also proposed. The proposed approach is easy to implement. It is proved that both the approximate solution and its partial derivatives converge to their exact solutions. Numerical results demonstrate that the proposed approach is effective and can provide a high precision global approximate solution.

Stochastic fractional differential equations are among the most significant and recent equations in physical mathematics. Consequently, several scholars have recently been interested in these equations to develop analytical approximations. In this study, we highlight the stochastic fractional space Allen-Cahn equation (SFACE) as a major application of this class. In addition, we utilize the simplest equation method (SEM) with a dual sense of Brownian motion to convert the presented equation into an ordinary differential equation (ODE) and apply an effective computational technique to obtain exact solutions. By carefully comparing the derived solutions with solutions from other articles, we prove the distinction of these solutions for their diversity and the discovery of new solutions for SFACE that appear in many scientific fields, such as mathematical biology, quantum mechanics, and plasma physics. The results introduced in this article were obtained by plotting several graphs and examining how noise affects exact solutions using Mathematica and MATLAB software packages.