System Of Hyperbolic Equations Research Articles

Nonclassical potential symmetry analysis and exact solutions for a thin film model of a perfectly soluble anti-surfactant solution

In this article, we construct exact solutions to a quasilinear system of hyperbolic partial differential equations which governs the dynamics of a thin film of a perfectly soluble anti-surfactant solution. The symmetry analysis for this system is performed first time in the literature to the best of our knowledge. In fact, we present a detailed and comprehensive study of symmetry analysis for the governing system and compute classical symmetries, nonclassical symmetries, nonlocal symmetries and nonclassical potential symmetries. Nonclassical potential symmetries appear to be very much useful in terms of obtaining several new hidden solutions of the system that can not be established by using classical symmetry reductions, nonclassical symmetry method, or potential symmetry analysis. By using the direct multipliers we demonstrate several conserved quantities of the governing system those yield associated nonlocally related potential systems to the given system. Further, we prove that the given system admits a nonlocal symmetry that arises from the symmetry-based method and consequently we obtain a family of exact solutions. We analyze the physical interpretation of some of the obtained solutions which include various soliton type solutions such as kink type solitons, breather type solitons, multiple solitons, singular kink type solitons and etc. Lastly, as an application, we analyze the evolutionary property of characteristic shock, weak discontinuity and collision between them by using one of the obtained solutions.

A study of dual phage lag model of heat transfer for solidification of binary eutectic alloy

In this paper, we developed a generalized non-Fourier model of energy transfer for solidification of a binary eutectic system (Al-Cu alloy). The mathematical model is divided into three stages: stage I (liquid region), II (liquid and mushy region), and III (liquid, mushy and solid region). The analysis of the model has been done in dimensionless form. This model is a boundary value problem of the system of second-order hyperbolic partial differential equations for Stage I, the moving boundary problem of the System of second-order hyperbolic partial differential equations for Stage II and III. For the solution, we applied Laplace transform technique in Stage I and Legendre wavelet spectral Galerkin method in Stage II and III. The effect of unit-less parameters: Relaxation time, Predvoditelev numbers, dimensionless thermal conductivity, Fourier numbers, Stefan number, solid fraction on temperature, and position of moving interface are discussed in detail.

Discontinuous Galerkin discretization in time of systems of second-order nonlinear hyperbolic equations

In this paper we study the finite element approximation of systems of second-order nonlinear hyperbolic equations. The proposed numerical method combines ahp-version discontinuous Galerkin finite element approximation in the time direction with anH1(Ω)-conforming finite element approximation in the spatial variables. Error bounds at the temporal nodal points are derived under a weak restriction on the temporal step size in terms of the spatial mesh size. Numerical experiments are presented to verify the theoretical results.

Similarity solution for isothermal flow behind the magnetogasdynamic cylindrical shock wave in a rotating non-ideal gas with the effect of the gravitational field

In this paper, we investigate a system of quasilinear hyperbolic partial differential equations, which describes the propagation of cylindrical shock waves in a rotating non-ideal gas with the effects of the gravitational field and the axial magnetic field. It is assumed that the flow is isothermal. The Lie group of transformations is used to generate the self-similar solutions of the considered problem in the medium of uniform density. The axial and azimuthal components of fluid velocity and magnetic field are supposed to be varying. We find the generators of the Lie group of transformations by employing the invariant surface criteria. We discovered four alternative solutions by selecting the arbitrary constants indicated in the generators' phrase. Only in three out of these four cases, the self-similar solutions exist. Two types of shock paths appear while solving the above cases. The power-law shock path appears in the first and third cases, while the exponential-law shock path appears in the second case. To find self-similar solutions, these cases have been solved numerically. We graphically show the distributions of flow variables behind the shock wave so that we can observe the effect on flow variables of the various values of the non-ideal parameter, Alfvén Mach number, adiabatic exponent, gravitational parameter, and ambient azimuthal velocity exponent. For the computational task, we used “MATLAB” software.

Solution of a nonlocal problem for hyperbolic equations with piecewise constant argument of generalized type

This paper proposes an effective method for solving a nonlocal problem for a system of second-order hyperbolic equations with piecewise constant time argument of generalized type. The method is based on the introduction of functional parameters that are set as the values of the desired solution along the lines of the domain partition with respect to the time variable. With the aid of the functional parameters and new unknown functions, the considered problem is reduced to an equivalent problem for a system of hyperbolic equations with data on the interior partition lines and functional relations with respect to the introduced parameters. We have developed a two-stage procedure to approximately solve the latter problem: firstly, we solve an initial-value problem for a system of differential equations in functional parameters; then, we solve a problem for a system of hyperbolic equations in new unknown functions with data on the interior partition lines. We have obtained some conditions for the convergence of approximate solutions to the exact solution of the problem under study in terms of input data and proved that these conditions guarantee the existence of a unique solution of the equivalent problem. Finally, we have established coefficient conditions for the unique solvability of the nonlocal problem.

Local characteristic decomposition based central-upwind scheme

We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the studied systems come from the complicated wave structures, such as shocks, rarefactions and contact discontinuities, arising even for smooth initial conditions. In order to reduce the diffusion in the original central-upwind schemes, we use a local characteristic decomposition procedure to develop a new class of central-upwind schemes. We apply the developed schemes to the one- and two-dimensional Euler equations of gas dynamics to illustrate the performance on a variety of examples. The obtained numerical results clearly demonstrate that the proposed new schemes outperform the original central-upwind schemes.

Fluid transients in viscoelastic pipes via an internal variable constitutive theory

This work presents a fluid transient model capable of handling the viscoelastic behavior of the pipe. A previously developed quasi-2D flow model is employed as a base, and the viscoelastic behavior of the pipe is incorporated by considering constitutive equations formulated in a thermodynamically consistent framework of an internal variable theory. Such an approach straightforwardly provides expressions for computing the rates of energy dissipation in the fluid and pipe accurately and separately. This novel feature discerns the local and overall impacts of the energy dissipation on the pressure oscillations caused by each medium. The governing equations of the model form a hyperbolic system of partial differential equations whose approximated solutions are obtained by the method of characteristics. Taking as reference pressure signals obtained by a classic reservoir-pipe-valve experiment found in the literature, it is shown that the model predictions are fully consistent. A comparison between the pressure responses of viscoelastic and elastic pipes reveals that in addition to delaying the pressure oscillations, the viscoelastic behavior causes a faster attenuation of them. The rates of energy dissipation in the viscoelastic pipe attain significant magnitudes during the first moments of the fluid transient and alters the hydrodynamical behavior of the flow. Such interference is exposed by comparing the responses of the same experimental setup when two different viscoelastic pipe materials are considered. It is also shown that the knowledge of the parcels of energy dissipated in the fluid and pipe individually can improve the comprehension of the phenomenon and be utilized for theoretical and applied research in the field.

Nonlinear fractional diffusion model for space-time neutron dynamics

In this work, a new fractional multidimensional neutron diffusion model is introduced using Cattaneo equation instead of the commonly used Fick's law for two neutron energy groups. The model represents a coupled system of nonlinear hyperbolic fractional partial differential equations considering finite fast and finite thermal neutron velocities. The model provides new quantities that relate some of the most important nuclear parameters together producing some newly physical concepts. These quantities include fast and thermal neutron relaxation times, Riemann-Liouville fractional integrals of the neutron densities and derivatives of macroscopic absorption cross sections. This model can be useful for mathematicians and physicists since the anomalous subdiffusion process describes the neutron movement closer to reality. The effect of the anomalous subdiffusion exponent (0<γ≤1) on the neutron flux is discussed. The model is successfully applied to simple homogenous and heterogonous reactor cores with step, ramp and sinusoidal perturbations for different values ofγ to estimate the average reactor powers in the fuel and moderator regions. Also, the new fractional model is applied to LRA BWR benchmark problem including adiabatic heating and Doppler feedback. On the other hand and based on the concept of prompt jump, the proposed fractional model is simplified and then analytically solved using Mittag-Liffler function. Solving fractional models numerically consumes considerable computational time costs due to the persistent memory of fractional operators, especially for smaller values of γ. For this regard, adaptively multi-step differential transform MDTM method is implemented to solve the proposed fractional model. The MDTM is developed using step size control technique for which the length of the step integration varies according to the solution behaviour.

Lie symmetry analysis for two-phase flow with mass transfer

Abstract We perform a complete symmetry classification for the hyperbolic system of partial differential equations, which describes a drift-flux two-phase flow in a one-dimensional pipe, with a mass-transfer term between the two different phases of the fluid. In addition, we consider the polytropic equation of states parameter and gravitational forces. For general values of the polytropic indices, we find that the fluid equations are invariant under the elements of a three-dimensional Lie algebra. However, additional Lie point symmetries follow for specific values of the polytropic indices. The one-dimensional systems are investigated in each case of the classification scheme, and the similarity transformations are calculated in order to reduce the fluid equations into a system of ordinary differential equations. Exact solutions are derived, while the reduced systems are studied numerically.

Multidimensional fully adaptive lattice Boltzmann methods with error control based on multiresolution analysis

Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fine mesh is unnecessary, still ensuring the overall quality of the numerical solution through error control. This work provides a possible answer to this interesting question, by connecting, for the first time, the field of lattice-Boltzmann Methods (LBM) to the adaptive multiresolution (MR) approach based on wavelets. To this end, we employ a MR multi-scale transform to adapt the mesh as the solution evolves in time according to its local regularity. The collision phase is not affected due to its inherent local nature and because we do not modify the speed of the sound, contrarily to most of the LBM/Adaptive Mesh Refinement (AMR) strategies proposed in the literature, thus preserving the original structure of any LBM scheme. Besides, an original use of the MR allows the scheme to resolve the proper physics by efficiently controlling the accuracy of the transport phase. We carefully test our method to conclude on its adaptability to a wide family of existing lattice Boltzmann schemes, treating both hyperbolic and parabolic systems of equations, thus being less problem-dependent than the AMR approaches, which have a hard time guaranteeing an effective control on the error. The ability of the method to yield a very efficient compression rate and thus a computational cost reduction for solutions involving localized structures with loss of regularity is also shown, while guaranteeing a precise control on the approximation error introduced by the spatial adaptation of the grid. The numerical strategy is implemented on a specific open-source platform called SAMURAI with a dedicated data-structure relying on set algebra.

On the theoretical foundation of overset grid methods for hyperbolic problems II: Entropy bounded formulations for nonlinear conservation laws

We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes a two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. In addition to the interface coupling, which is required, entropy dissipation and coupling can optionally be added through the use of linear penalties within the overlap region.

Converging strong shock wave from a cylindrical piston in a Van der Waals magnetogasdynamics with dust particles

In this study, by using the perturbation series approach, a global solution to the implosion problem is obtained for a quasi-linear hyperbolic system of partial differential equations (PDEs) describing a problem of strong converging cylindrical shock waves collapsing at the axis of symmetry in a Van der Waals dusty gas under the effect of axial magnetic field. This global solution provides the results of Guderley’s local self-similar solution which is valid only in the neighborhood of the axis of implosion. The similarity exponents and corresponding amplitudes are obtained in the neighborhood of the shock collapse. In addition, the values of leading similarity exponents are compared with the results obtained by Whitham’s method. The shock trajectory and flow variables (i.e., density, velocity, pressure and magnetic pressure) have been drawn for different values of the relative specific heat, the mass concentration of dust particles, the ratio of the density of solid particles to the initial density of the gas, Van der Waals excluded volume and shock Cowling number.

Study of wave propagation in arterial blood flow under symmetry analysis

In this paper, we address a one‐dimensional quasi‐linear hyperbolic system of equations describing blood flow through compliant axi‐symmetric vessels. From the symmetry analysis, we derive symmetry group of transformations and the corresponding symmetry generators by analyzing the parameters. Next, with the help of symmetry generators and invariant functions, we construct and classify the optimal system of subalgebras. Further, we obtained the similarity variables and similarity forms for each subalgebra leading to the reduction of the given governing coupled PDEs to the system of ODEs. Moreover, we studied the nature of blood flow velocity as well as the cross‐sectional area of the arteries under the influence of arterial stiffness graphically. Finally, the evolutionary behavior of weak discontinuity in the blood flow pattern is discussed with respect to aging.

Application of the Generalized Method of Moving Coordinates to Calculating Stress Fields near an Elliptical Hole.

The distribution of stresses near holes is of great importance in fracture mechanics and material modeling. The present paper provides a general stress solution near a traction-free surface for an arbitrary piecewise linear yield criterion, assuming plane-strain conditions. The generalized method of moving coordinates is proven efficient in this case. In particular, the solution reduces to evaluating one ordinary integral. The boundary value problem solved is a Cauchy problem for a hyperbolic system of equations. Therefore, the stress solution in the plastic region is independent of other boundary conditions, though the occurrence of plastic yielding at a specific point is path-dependent. The general solution applies to calculating the stress field near an elliptic hole. It is shown that the parameter that controls the pressure-dependency of the yield criterion affects the stress field significantly. The aspect ratio is less significant as compared to that parameter. However, for a given material, the aspect ratio should also be considered to predict the stress field accurately, especially in the near vicinity of the hole. The solution reduces to an available solution for the pressure-independent yield criterion, which is a particular yield criterion of the considered class of yield criteria.

On a class of systems of hyperbolic equations describing pseudo-spherical or spherical surfaces

We consider systems of partial differential equations of the form{uxt=F(u,ux,v,vx),vxt=G(u,ux,v,vx), describing pseudospherical (pss) or spherical surfaces (ss), meaning that, their generic solutions (u(x,t),v(x,t)) provide metrics, with coordinates (x,t), on open subsets of the plane, with constant curvature K=−1 or K=1. These systems can be described as the integrability conditions of g-valued linear problems, with g=sl(2,R) or g=su(2), when K=−1, K=1, respectively. We obtain characterization and also classification results. Applications of the theory provide new examples and new families of systems of differential equations, which also contain generalizations of a Pohlmeyer-Lund-Regge type system and of the Konno-Oono coupled dispersionless system. We provide explicitly the first few conservation laws, from an infinite sequence, for some of the systems describing pss.

Mathematical Analysis of Dynamic Models of Suspension Bridges with Delayed Damping

Suspension bridges are a type of construction in which the deck is suspended under a series of suspension cables that are on vertical hangers. The first modern example of this project began to appear in the early 1800s. Modern suspension bridges are lightweight, aesthetically pleasing and can span longer distances than any other bridge form. Many papers have been devoted to the modelling of suspension bridges, for instance, Lazer and McKenna studied the problem of nonlinear oscillation in a suspension bridge. They introduced a (one-dimensional) mathematical model for the bridge that takes into account of the fact that the coupling provided by the stays connecting the main cable to the deck of the road bed is fundamentally nonlinear, that is, they gave rise to the system of semi linear hyperbolic equation, where the first equation describes the vibration of the road bed in the vertical plain and the second equation describes that of the main cable from which the road bed is suspended by the tie cables. Recently, interest in this field has been increasing at a high rate. In this paper, we investigate some mathematical models of suspension bridges with a strong delay in linear aerodynamic resistance force. We establish the exponential decay of the solution for the corresponding homogeneous system and prove the existence of an absorbing set as well as a bounded attractor.

Power Arcing Fault Detection and Location Algorithm Solution in Two and Three Dimensions Euclidean Planes Using an Analytic Geometry Method Instead of an Iterative or Numerical Method

The relationship between the electric current and the magnetism is well demonstrated by the Faraday's induction and conductivity laws. Under this concept, it follows that, through intensive laboratory experiments, it has been established that the arcing fault current path related sparks are indeed source of ElectroMagnetic Radiated (EMR) signals waves. These EMR can be detected and located at a relatively remote distance, with the aid of directional antennas arrays, at considerable and reasonable accuracy. Under this novel concept of power arcing fault source point triangulation, the arc EMR data play a critical role in providing essential timing information onto which the arc fault detection and location algorithm is based. Thus, from these arc EMR recorded and gathered data, the signals Time Differences Of Arrival (TDOA) information or signatures can be easily extracted from a pair of antennas, with the aid of an appropriate signal processing tool, namely the cross-correlation (Xcorr) function. Since the electric arc phenomenon is unpredictable practically, therefore, the use of the arc signal Time Of Arrival (TOA), is impossible for the fault source point triangulation solution. In contrast, the values of TDOA seem to be efficient for solving the system of hyperbolic equations which has been imposed by this particular field measurement structure and constraints. It is of interest to mention that, the arc fault EMR field measurement structure and constraints are given and determined by the distributed antennas arrays locations or positions only. In other words, with the knowledge of both the antennas arrays locations Cartesian Coordinates (CC), together with the TDOA values, it is possible to pinpoint the unknown target power arcing fault exact source point position or location, not only in two-dimensions (2D), but also in three-dimensions (3D) Euclidean plane. Thus, this kind of arc fault source point position or location solution finding is generally referred to as a source point triangulation method. This source point triangulation method consists of solving a system of hyperbolic equations as aforementioned. In fact, under a geometry analysis process, it follows that, if only if the arcing fault hazard formation time is known in advance, therefore, the solution of a system of hyperbolic equations consists of finding the intersection point of at least two or more circles. Such that, each of these circles, will be centered at each antenna in such a way that each circle will have a radius given by the distance derived from the value of TOA together with the power arcing fault hazard ignition time stamp or signature. In contrast, since the arcing fault hazard formation stage time is unknown in practice, therefore the solution of the system of hyperbolic equations consists of finding the intersection point of at least two or more hyperbolas. Such that each of these hyperbolas will have a focal point determined by the location of each antenna and the distance between pair of foci will be given by the values of TDOA. However, generally, a system of hyperbolic equations can be solved either with the aid of an analytic method or with that of a numerical method (also known as iterative method). Unlike the majority of this research field related relevant published literature into which the iterative methods have been used intensively, this paper presents an accurate analytic geometry method in order to solve the electric arc faults EMR source point exact location or position. As a result, it appears that, this novel proposed analytic geometry method has shown its potential in clarifying the power arcing fault source position or location, at a reasonable and considerable accuracy.

A new model for fluid transients in piping systems taking into account the fluid–structure interaction

The present work extends a recently developed quasi-2D flow model for fluid transients in elastic pipes to accommodate fluid–structure interaction mechanisms. In this context, the primary goal of the proposed approach is to analyze energy transfer and dissipative effects in the fluid–pipe system. The fluid–structure interaction couples the flow dynamics with the axial movement of the tube, giving rise to friction, Poisson, and junction coupling mechanisms to take place. The mechanical model mathematical structure forms a quasi-linear hyperbolic system of partial differential equations for which approximated solutions are sought by employing the method of characteristics. The proposed model reproduces classical benchmark solutions and presents a good agreement with experimental data. In addition, the whole thermomechanical consistent framework in which the model is established allows an accurate description of the flow’s internal structure. This feature allows a better comprehension of the phenomenon when compared to the classic frictionless and quasi-steady-friction-based four-equation models. The present approach allows a better description of the friction coupling mechanism and unveils uneven shear stress distributions in the unsteady flow. As a result, the energy dissipation in the fluid is captured with accuracy. Due to the absence or limited ability to describe these dispersive and dissipative effects, the traditional approaches are proved to lose their accuracy as the fluid transient goes. The coupled and uncoupled models are also analyzed and are proven to respond differently due to localized pipe–fluid interface effects in addition to the bulk mechanisms of transfer of energy that occur differently in both approaches.

On the Darboux Problem for a Hyperbolic System of Equations with Multiple Characteristics

On the Darboux Problem for a Hyperbolic System of Equations with Multiple Characteristics

Simulation of flow in an artery under pathological hemodynamic conditions: The use of a diagnostic disease descriptor

A numerical model for simulating and predicting blood flow dynamics in diseased arterial vessels has been developed. The time-dependent one-dimensional hyperbolic system of quasilinear partial differential equations which incorporates a diagnostic disease descriptor (kD) was used to simulate transient flow distribution for idealized healthy and diseased states. Blood flow simulations in the iliac arteries over about 125% of a cardiac cycle were generated and calibrated using the kD values from 0 to 3 representing hypothetical diseased states. Early results indicate that disease conditions induce abnormal flow in the artery, generating disorder and increased amplitude of blood pressure, flow and distensibility with increasing numerical values of the disease factor kD. More so, the prospective use of the kD-approach with documentation of in vivo adverse flow visualizations for diagnostic purposes was decisively discussed.