Mixed Initial Conditions Research Articles

A modified fuzzy Adomian decomposition method for solving time-fuzzy fractional partial differential equations with initial and boundary conditions

This research article introduces a novel approach based on the fuzzy Adomian decomposition method (FADM) to solve specific time fuzzy fractional partial differential equations with initial and boundary conditions (IBCs). The proposed approach addresses the challenge of incorporating both initial and boundary conditions into the FADM framework by employing a modified approach. This approach iteratively generates a new initial solution using the decomposition method. The method presented here offers a significant contribution to solving fuzzy fractional partial differential equations (FFPDEs) with fuzzy IBCs, a topic that has received limited attention in the literature. Furthermore, it satisfies a high convergence rate with minimal computational complexity, establishing a novel aspect of this research. By providing a series solution with a small number of recursive formulas, this method enhances accuracy and emerges as a preferred choice for tackling FFPDEs with mixed initial and boundary conditions. The effectiveness of the proposed technique is further supported by the inclusion of several illustrative examples.

Pseudo-almost periodic $C^{0}$-solution for evolution inclusion with mixed nonlocal plus local initial conditions

This paper is devoted to the study of a class of evolution inclusion in Banach spaces with nonlocal plus local mixed initial conditions. Under some mild assumptions, a unique solvability result to the multivalued evolution problem is obtained via the arguments of fixed point principle and the theory of $C^0$-semigroup.

Order and Disorder in a Cyclically Competitive Ecological Community

We consider a deterministic model of a rock-paper-scissors cyclic ecological system. The model accounts for species mobility via Fickian diffusion, as well as interspecies interactions describing the cyclic competition scheme. The system admits both single-species equilibria (saddles) and a three-species coexistence equilibrium state. We focus on the regime where coexistence is unstable. When there is no spatial dependence, the solution of the corresponding ordinary differential equation system admits an attracting heteroclinic cycle consisting of orbits connecting the three single-species saddles in the sequence, to which most solutions with physical initial conditions asymptote. When there is spatial dependence (partial differential equation system), we consider mixed initial conditions (all species cohabit the same spatial region), as well as patch initial conditions, where each species is isolated from the other two, and show that, when interspecies competition is not strong, spatiotemporal chaotic behavior generally occurs. We propose a mechanism for the development of chaos—patch splitting—whereby sufficiently large patches are repeatedly split by the diffusion of competitors into the patch. However, when there is strong interspecies competition, ordered spatiotemporal patterns can occur. We consider both 1D patterns, corresponding to a community confined to a thin, circular annulus, and 2D geometries, simulating a petri dish. We show that, in 1 dimension, traveling arrays of single-species patches, as well as modulated traveling waves, consisting of patches which periodically expand and contract (breather modes), exist. In 2 dimensions, spirals, as well as localized patches chasing each other in the sequence, can occur. We consider length and temporal scales appropriate for a bacterial community in a laboratory setting, qualitatively modeling observed Escherichia coli cyclic bacterial systems.

Necessary optimality conditions of delay parameters for the nonlinear optimization problem with the mixed initial condition

Necessary optimality conditions of delay parameters for the nonlinear optimization problem with the mixed initial condition

Numerical study of the particles segregation phenomenon in the fluidized beds

Purpose The purpose of this paper is to present a computational fluid dynamic simulation for the investigation of the particles segregation phenomenon in the gas–solid fluidized beds. Design/methodology/approach These particles have the same size and different densities. The k–ε model and multiphase particle-in-cell method have been utilized for modeling the turbulent fluid flow and solid particles behaviors, respectively. The coupled equations of the velocity and pressure have been solved by using a combination of SIMPLE and PISO algorithms. After validating the simulation, different mixing indices, with different calculation bases, have been investigated, and it has been found that the Lacey mixing index, which was defined based on statistical concepts, is suitable to investigate the segregation/mixing phenomena of this bed in different conditions. Finally, the effects of parameters such as velocity, particle density ratio, jetsam concentration, and initial arrangement on the segregation/mixing behaviors of the bed have been studied. Findings The results show that the increase in the superficial gas velocity decreases the mixing index to a minimum value and then increases this index in the beds with mixed initial condition, unlike the beds with separated initial condition. Moreover, an increase in the particle density ratio increases the minimum fluidization velocity of the bed, and also the amount of segregation, and increase in the jetsam concentration increases the value of the mixing index. Originality/value A computational fluid dynamics simulation has been presented for the particles segregation phenomenon in the gas–solid fluidized beds.

Fractional Retarded Differential Equations Involving Mixed Nonlocal Plus Local Initial Conditions

The aim of this paper is to discuss the existence of mild solutions and positive mild solutions for a general class of semilinear fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions on infinite dimensional Banach spaces. Under the situation that the nonlinear term and nonlocal function satisfy some appropriate growth conditions and a noncompactness measure condition, we obtained the existence of mild solutions and positive mild solutions by utilizing a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness. In addition, the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example about the retarded parabolic partial differential equation involving a general mixed nonlocal plus local initial conditions is also given to illustrate the feasibility of our abstract results.

The Impacts of Using Mixed Physics in the Community Leveraged Unified Ensemble

Abstract A well-known problem in high-resolution ensembles has been a lack of sufficient spread among members. Modelers often have used mixed physics to increase spread, but this can introduce problems including computational expense, clustering of members, and members that are not all equally skillful. Thus, a detailed examination of the impacts of using mixed physics is important. The present study uses two years of Community Leveraged Unified Ensemble (CLUE) output to isolate the impact of mixed physics in 36-h forecasts made using a convection-permitting ensemble with 3-km horizontal grid spacing. One 10-member subset of the CLUE used only perturbed initial conditions (ICs) and lateral boundary conditions (LBCs) while another 10-member ensemble used the same mixed ICs and LBCs but also introduced mixed physics. The cases examined occurred during NOAA’s Hazardous Weather Testbed Spring Forecast Experiments in 2016 and 2017. Traditional gridpoint metrics applied to each member and the ensemble as a whole, along with object-based verification statistics for all members, were computed for composite reflectivity and 1- and 3-h accumulated precipitation using the Model Evaluation Tools (MET) software package. It is found that the mixed physics increases variability substantially among the ensemble members, more so for reflectivity than precipitation, such that the envelope of members is more likely to encompass the observations. However, the increased variability is mostly due to the introduction of both substantial high biases in members using one microphysical scheme, and low biases in other schemes. Overall ensemble skill is not substantially different from the ensemble using a single physics package.

Effect of pressure-dilatation on energy spectrum evolution in compressible turbulence

The effect of internal-kinetic energy exchange on transient spectral energy transfer in compressible turbulence is investigated. We derive the spectral evolution equations for kinetic energy and pressure fields to highlight the key mechanisms that affect the turbulence spectral evolution. Direct numerical simulations of decaying isotropic turbulence are performed from solenoidal, dilatational, and mixed velocity initial conditions. It is shown that internal-kinetic energy exchange arising due to pressure-dilatation renders the dilatational kinetic energy amplitudes at large scales of motion to be oscillatory. The oscillatory behavior of amplitude diminishes with increasing wavenumber. The dilatational energy spectrum also exhibits a wider range of scales due to its inherent tendency to form shocks. The findings are expected to lead to an improved understanding of energy dynamics in high-speed compressible flows.

Shelf Cross-Shore Flows under Storm-Driven Conditions: Role of Stratification, Shoreline Orientation, and Bathymetry

AbstractNumerical simulations are used to study the response of Long Bay, South Carolina, a typical coastal embayment with curved coastline located on the South Atlantic Bight, to realistic, climatologically defined, synoptic storm forcing. Synoptic storms, consisting of cold and warm fronts as well as tropical storms, are used as forcing under both mixed and stratified initial conditions. The analysis focuses on the development of cross-shore shelf circulation and the relative contributions of regionally defined cross-shore winds and alongshore bathymetric variation. The simulation results show that, under stratified conditions, the regionally defined offshore-directed wind component promotes upwelling during the developing stage of the cold front and enhances mixing during the decaying stage. No significant effect is found for warm front and tropical storm forcing conditions. Net cross-shore transports are induced at the southern and northern sides of the embayment that have opposing signs. Besides the surface and bottom Ekman transports, geostrophic transport due to alongshore shelf bed slope and horizontal advection are found to be important contributors to cross-shore flow development. Sea level variability along the curved coastline is driven by the regional alongshore wind, but a spatial variability is identified from the locally defined components of along- and cross-shore winds controlled by coastline orientation.

A Reliable Algorithm for Multi-Dimensional Mobile/Immobile Advection-Dispersion Equation with Variable Order Fractional

Objectives: Presented a modified treatment of initial boundary value problems for multi-dimensional mobile/immobile advection-dispersion equation with variable order fractional (MDMADEVF). Methods: we applied mixed initial and boundary conditions together using variational iteration method (VIM) to get a new initial solution at every iteration. Findings: The simulation results show the proposed system accurate and the utilized algorithm simple and easy to implement. Application: numerical examples were provided to show that variational iteration method was computationally efficient. The results were presented in tables and figures using the MathCAD 12 and Matlab software package when it is needed. Keywords: Caputo Derivatives, Fractional Mobile / Immobile Advection-Dispersion Model

Numerical solution of high order linear complex differential equations via complex operational matrix method

In this work, a numerical method based on a complex operational matrix is utilized to solve high order linear complex differential equations under mixed initial conditions. For this aim, we introduce orthonormal Bernstein polynomials (OBPs), and we obtain their complex operational matrix of differentiation. The main advantage of the proposed method is that by using this method complex differential equations reduce to a linear system of algebraic equations which can be solved by using an appropriate iterative method. To, some results concerning the error analysis associated with the present method are discussed. Finally, we give some numerical examples to reveal accuracy and efficiency of the proposed method. Also, the numerical results obtained by this method are compared with numerical results achieved from other existing methods.

Transient reaction of an elastic half-plane on a source of a concentrated boundary disturbance

One of the key problems in studying the non-stationary processes of solid mechanics is obtaining of influence functions. These functions serve as solutions for the problems of effect of sudden concentrated loads on a body with linear elastic properties. Knowledge of the influence functions allows us to obtain the solutions for the problems with non-mixed boundary and initial conditions in the form of quadrature formulae with the help of superposition principle, as well as get the integral governing equations for the problems with mixed boundary and initial conditions. This paper offers explicit derivations for all nonstationary surface influence functions of an elastic half-plane in a plane strain condition. It is achieved with the help of combined inverse transform of a Fourier-Laplace integral transformation. The external disturbance is both dynamic and kinematic. The derived functions in xτ-domain are studied to find and describe singularities and are supplemented with graphs.

On the Existence of Strong Solutions for a Degenerate Parabolic Inequality with Mixed Boundary Conditions

The degenerate parabolic variational inequality with mixed boundary conditions and inhomogeneous initial conditions is studied in the case where the corresponding operator can lose the properties of coercivity and continuity in the corresponding Sobolev spaces. By using the Hardy–Poincare inequality, we prove the unique solvability of the original evolutionary variational inequality under the condition that the degenerate weight function is a function of potential type.

Dynamics of entanglement and quantum discord in the Tavis–Cummings model

We revisit the problem of the dynamics of quantum correlations in the Tavis–Cummings model. Our results show that the dynamics of entanglement and quantum discord are far from being trivial or intuitive. We find states with the same entanglement but different discord and states where the two quantifiers give opposite information about correlations at a certain time. We furthermore show that many of the dynamical features of quantum discord attributed to dissipation are already present in the exact framework and are due to the characteristic quantum nonlinearity of the model and to the choice of initial conditions. Through a comprehensive analysis of pure and mixed initial conditions, we find a fascinating range of phenomena that can be used for experimental purposes. We propose an experiment called quantum discord gates where for a given pure initial condition discord is zero or non-zero depending on the number of photons in the cavity. Given the marginal character of states with zero discord this result is not only completely counterintuitive but is also useful as a way to count photons.

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.

Delay evolution equations with mixed nonlocal plus local initial conditions

We consider the delay differential equation u′(t) ∈ Au(t) + f(t, ut), t ∈ ℝ+, where A is the infinitesimal generator of a nonlinear semigroup of contractions in a Banach space X and f is continuous, subjected to a general mixed nonlocal + local initial condition of the form u(t) = g(u)(t) + ψ(t), t ∈ [-τ, 0]. We prove that under natural conditions on A, f, g and ψ the problem above has at least one C0-solution. Applications to T-periodic and to T-anti-periodic problems, as well as an illustrative example referring to the nonlinear diffusion equation are also included.

Structural Stability of a Dynamical System Near a Non-Hyperbolic Fixed Point

We prove structural stability under perturbations for a class of discrete-time dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well-posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. The class of dynamical systems we study, and the boundary conditions we impose, arise in a renormalisation group analysis of the 4-dimensional weakly self-avoiding walk and the 4-dimensional n-component $|\varphi|^4$ spin model.

A new procedure of initial boundary value problems using homotopy perturbation method

In this paper, , a new procedure is applied to treatment of initial boundary value problems by mixed initial and boundary conditions together  to obtain a new initial solution at every iteration using homotopy perturbation method. The structure of a new successive initial solutions can give a more accurate solution. Some examples (linear and nonlinear, homogeneous and nonhomogeneous problems) are given in this paper to illustrate the effectiveness and convenience of a new procedure.

Cold dark matter isocurvature perturbations: Cosmological constraints and applications

In this paper we present the constraints on cold dark matter (CDM) isocurvature contributions to the cosmological perturbations. By employing Markov chain Monte Carlo method, we perform a global analysis for cosmological parameters using the latest astronomical data, such as 7-year Wilkinson Microwave Anisotropy Probe (WMAP7) observations, matter power spectrum from the Sloan Digital Sky Survey, luminous red galaxies, and the Union2 type Ia supernovae sample. We find that the correlated mixture of adiabatic and isocurvature modes is slightly favored by the current observational data. We also obtain a tight limit on the fraction of the CDM isocurvature contributions, which should be less than 14.6% at 95% confidence level. With the presence of the isocurvature modes, the adiabatic spectral index becomes slightly bigger, ${n}_{s}^{\mathrm{adi}}=0.972\ifmmode\pm\else\textpm\fi{}0.014$ ($1\ensuremath{\sigma}$), and the tilt for isocurvature spectrum could be large, namely, the best fit value is ${n}_{s}^{\mathrm{iso}}=3.020$. Finally, we discuss the effect on WMAP normalization priors, shift parameter $R$, acoustic scale ${l}_{A}$, and ${z}_{*}$, from the CDM isocurvature perturbation. By fitting the mixed initial condition to the combined data, we find the mean values of $R$, ${l}_{A}$, and ${z}_{*}$ can be changed about $2.9\ensuremath{\sigma}$, $2.8\ensuremath{\sigma}$, and $1.5\ensuremath{\sigma}$ respectively, comparing with those obtained in the pure adiabatic condition.

Lax–Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton–Jacobi Equation. Part I: Theory

This article proposes a new approach for computing a semi-explicit form of the solution to a class of Hamilton-Jacobi (HJ) partial differential equations (PDEs), using control techniques based on viability theory. We characterize the epigraph of the value function solving the HJ PDE as a capture basin of a target through an auxiliary dynamical system, called ?characteristic system?. The properties of capture basins enable us to define components as building blocks of the solution to the HJ PDE in the Barron/Jensen-Frankowska sense. These components can encode initial conditions, boundary conditions, and internal ?boundary? conditions, which are the topic of this article. A generalized Lax-Hopf formula is derived, and enables us to formulate the necessary and sufficient conditions for a mixed initial and boundary conditions problem with multiple internal boundary conditions to be well posed. We illustrate the capabilities of the method with a data assimilation problem for reconstruction of highway traffic flow using Lagrangian measurements generated from Next Generation Simulation (NGSIM) traffic data.