Mixed Equations Research Articles

Projection-iterated method for solving numerically the nonlinear mixed integral equation in position and time

In this work, we consider a nonlinear Volterra-Hammerstein integral equation of the second kind (V-HIESK). The existence of a unique solution to the integral equation, under certain conditions, is guaranteed. The projection method was used, as the best numerical method. This method is better than the previous methods in the possibility of obtaining the lowest relative error. The method of obtaining the best approximate solution, through the projection method, depends on presenting three consecutive algorithms and depends mainly on the method of iterative projection (P-IM). In each algorithm, we get the approximate solution and the corresponding relative error. Moreover, we demonstrated that the estimated error of P-IM in the first algorithm is better than that of the successive approximation method (SAM). Some numerical results were calculated, and the error estimate was computed, in each case. Finally, the numerical results have been compared between this method and previous research, and it is clear through the same examples that the relative error of the first algorithm is better than the comparative error in the other methods.

A nonlocal problem for a mixed equation of high even order with a fractional Caputo derivative

A nonlocal problem for a mixed equation of high even order with a fractional Caputo derivative

Reliabilities of estimated breeding values in models with metafounders

BackgroundReliabilities of best linear unbiased predictions (BLUP) of breeding values are defined as the squared correlation between true and estimated breeding values and are helpful in assessing risk and genetic gain. Reliabilities can be computed from the prediction error variances for models with a single base population but are undefined for models that include several base populations and when unknown parent groups are modeled as fixed effects. In such a case, the use of metafounders in principle enables reliabilities to be derived.MethodsWe propose to compute the reliability of the contrast of an individual’s estimated breeding value with that of a metafounder based on the prediction error variances of the individual and the metafounder, their prediction error covariance, and their genetic relationship. Computation of the required terms demands only little extra work once the sparse inverse of the mixed model equations is obtained, or they can be approximated. This also allows the reliabilities of the metafounders to be obtained. We studied the reliabilities for both BLUP and single-step genomic BLUP (ssGBLUP), using several definitions of reliability in a large dataset with 1,961,687 dairy sheep and rams, most of which had phenotypes and among which 27,000 rams were genotyped with a 50K single nucleotide polymorphism (SNP) chip. There were 23 metafounders with progeny sizes between 100,000 and 2000 individuals.ResultsIn models with metafounders, directly using the prediction error variance instead of the contrast with a metafounder leads to artificially low reliabilities because they refer to a population with maximum heterozygosity. When only one metafounder is fitted in the model, the reliability of the contrast is shown to be equivalent to the reliability of the individual in a model without metafounders. When there are several metafounders in the model, using a contrast with the oldest metafounder yields reliabilities that are on a meaningful scale and very close to reliabilities obtained from models without metafounders. The reliabilities using contrasts with ssGBLUP also resulted in meaningful values.ConclusionsThis work provides a general method to obtain reliabilities for both BLUP and ssGBLUP when several base populations are included through metafounders.

Plane problems of multiple interactions of rigid punches and bodies with complex multilayer coatings

The article deals with the most general formulation of the problem of the interaction of a system of rigid punches and a viscoelastic aging foundation covered by several different nonuniform thin layers of variable thickness. It is shown that the mathematical model of such a problem is a system of mixed integral equations with additional integral conditions. On the basis of a special approach described in present article, an analytical solution is obtained in series for various versions of the problem statement. It is indicated that the obtained solution representation is effective even in the case when the properties of the coating layers and the shape of the contacting surfaces are described by complex functions.

Discrete and continuous models of linear elasticity: history and connections

This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed differential-difference equation of a one-dimensional (1D) lattice and the partial differential equation of the associated continuum. A consistent three-dimensional (3D) generalization of this model was given much later: Poincaré and Voigt reconciled the molecular and the continuum approaches at the end of the nineteenth century, but only in 1912 Born and von Kármán presented the mixed differential-difference equations of discrete isotropic elasticity. Their model is a 3D generalization of Lagrange’s 1D lattice and considers longitudinal, diagonal and shear elastic springs among particles, so the associated continuum is characterized by three elastic constants. Born and von Kármán proved that the lattice equations converge to Navier’s partial differential ones asymptotically, thus being a formulation of continuous elasticity in terms of spatial finite differences, as for Lagrange’s 1D lattice. Neglecting shear springs in Born–Kármán’s lattice equals to Navier’s assumption of pure central forces among molecules: in the limit, the lattice behaves as a one-parameter isotropic solid (“rari-constant” theory: equal Lamé parameters, or, equivalently, Poisson’s ratio $$\upsilon =1/4$$ ). Hrennikoff and McHenry revisited the lattice approach with pure central interactions using a plane truss; the equivalent Born–Kármán’s lattice in plane stress in the limit tends to a continuum with Poisson’s ratio $$\upsilon = 1/3$$ . Contrary to McHenry–Hrennikoff’s truss, Born–Kármán’s lattice leads to a “free” Poisson’s ratio bounded by its “limit’ bound ( $$\upsilon =1/4$$ for plane strain or 3D elasticity; $$\upsilon =1/3$$ for plane stress elasticity). Unfortunately, Born–Kármán’s lattice model does not comply with rotational invariance principle, for non-central forces. The consistent generalization of Lagrange’s lattice in 3D was achieved only by Gazis et al. considering an elastic energy that depends on changes in both lengths and angles of the lattice. An alternative consistent three-parameter elastic lattice is the Hrennikoff’s, with additional structure in the cell. We also discuss the capability of nonlocal continuous models to bridge the gap between continuum isotropic elasticity at low frequencies and lattice anisotropic elasticity at high frequencies.

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

This paper deals with the solution of a third kind mixed integro-differential equation (MIDE) in displacement type in space L 2 − 1 , 1 × C 0 , T , T < 1 . The singular kernel is modified to take a logarithmic form, while the kernels of time are continuous and positive functions. Using the separation of variables technique, we have a system of Fredholm integral equations (FIEs) that can be transformed into an algebraic system after using orthogonal polynomials. In all the previous researchers’ works, the time periods were divided, and the mixed equation transformed into an algebraic system of FIEs. While when using the separation method, we are able to obtain FIE with time coefficients, and these functions are described as an integral operator in time. Thus, we can study the behavior of the solution with the time dimension in a broader and deeper than the previous one. Some examples are given and discussed to show the performance and efficiency of the proposed methods.

Legendre-Gauss-Lobatto collocation method for solving multi-dimensional systems of mixed Volterra-Fredholm integral equations

<abstract><p>Integral equations play a crucial role in many scientific and engineering problems, though solving them is often challenging. This paper addresses the solution of multi-dimensional systems of mixed Volterra-Fredholm integral equations (SMVF-IEs) by means of a Legendre-Gauss-Lobatto collocation method. The one-dimensional case is addressed first. Afterwards, the method is extended to two-dimensional linear and nonlinear SMVF-IEs. Several numerical examples reveal the effectiveness of the approach and show its superiority in comparison to other alternative techniques for treating SMVF-IEs.</p></abstract>

Physical phenomena of spectral relationships via quadratic third kind mixed integral equation with discontinuous kernel

<abstract> <p>Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an integral operator with a singular kernel. Second, using these relations to solve a mixed integral equation (<bold>MIE</bold>) of the second kind in the space $ {L}_{2}\left[-\mathrm{1, 1}\right]\times C\left[0, T\right], T < 1. $ The way to do this is to derive a general principal theorem of the spectral relations from the term of the Volterra-Fredholm integral equation (<bold>V-FIE</bold>), with the help of the Chebyshev polynomials (<bold>CPs</bold>), and then use the results in the general <bold>MIE</bold> to discuss its solution. More than that, some special and important cases will be devised that help explain many phenomena in the basic sciences in general. Here, the <bold>FI</bold> term is considered in position, in $ {L}_{2}\left[-\mathrm{1, 1}\right], $ and its kernel takes a logarithmic form multiplied by a general continuous function. While the <bold>VI</bold> term in time, in $ C\left[0, T\right], T < 1, $ and its kernels are smooth functions. Many numerical results are considered, and the estimated error is also established using Maple 2022.</p> </abstract>

Low-temperature vibronic spectroscopy of condensed chromophore exhibiting inhomogeneous distribution of vibrational frequencies in a mixed quantum-classical environment.

This work has been motivated by the recent paper by the author [M. Toutounji, Phys. Chem. Chem. Phys., 2021, 23, 21981] whereby a mixed quantum-classical Liouville equation was used to probe the spectroscopy and dynamics of a spin-boson system. A mixed quantum-classical Liouville equation treats the system of interest quantum mechanically, the bath classically, and the coupling term mixed quantum-classical mechanically. This paper offers a two-fold advantage: correcting the treatment of the electronic transition decay (width in frequency domain) and assessing the local heterogeneous vibrational structure. The homogeneous linear absorption spectrum of a chromophore embedded in a mixed quantum-classical environment at low temperature is composed of a sharp peak called a zero-phonon line (ZPL) and a broad phonon sideband (PSB), whereby the ZPL and the PSB are assimilated by a Lorentzian function and Voigt profiles, respectively. The PSB, in this case, is characterized by a local heterogeneous structure due to a dispersive medium of vibrations, modeled by vibrational Gaussian distributions to represent the arising inhomogeneous broadening and Lorentzians to model the homogeneous vibrations. This description seems to model proteins and amorphous solids exhibiting a local heterogeneous structure as both electronic and vibrational inhomogeneous broadening seems to be large in these media. This work provides a derivation of linear absorption lineshape and vibronic transition dipole moment time correlation functions, both of which account for pure electronic dephasing (ZPL width) the Voigt profile description of the phonon profiles (PSB) in dispersive media.

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

New algorithms for solving nonlinear mixed integral equations

<abstract><p>In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind is discussed. We also prove the solvability of the second kind of the NVFIE using the Banach fixed point theorem. Using quadrature method, the NVFIE leads to a system of nonlinear Fredholm integral equations (NFIEs). The existence and unique numerical solution of this system is discussed. Then, the modified Taylor's method was applied to transform the system of NFIEs into nonlinear algebraic systems (NAS). The existence and uniqueness of the nonlinear algebraic system's solution are discussed using Banach's fixed point theorem. Also, the stability of the modified error is presented. Some numerical examples are performed to show the efficiency and simplicity of the presented method, and all results are obtained using Wolfram Mathematica 11.</p></abstract>

A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order $ {r} \in (1, 2) $ with impulses

<abstract><p>In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order $ 1 < r < 2 $. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.</p></abstract>

Some results on mixed fractional integrodifferential equation in matrix MB-space

Some results on mixed fractional integrodifferential equation in matrix MB-space

A chaos-based adaptive equilibrium optimizer algorithm for solving global optimization problems.

The equilibrium optimizer (EO) algorithm is a newly developed physics-based optimization algorithm, which inspired by a mixed dynamic mass balance equation on a controlled fixed volume. The EO algorithm has a number of strengths, such as simple structure, easy implementation, few parameters and its effectiveness has been demonstrated on numerical optimization problems. However, the canonical EO still presents some drawbacks, such as poor balance between exploration and exploitation operation, tendency to get stuck in local optima and low convergence accuracy. To tackle these limitations, this paper proposes a new EO-based approach with an adaptive gbest-guided search mechanism and a chaos mechanism (called a chaos-based adaptive equilibrium optimizer algorithm (ACEO)). Firstly, an adaptive gbest-guided mechanism is injected to enrich the population diversity and expand the search range. Next, the chaos mechanism is incorporated to enable the algorithm to escape from the local optima. The effectiveness of the developed ACEO is demonstrated on 23 classical benchmark functions, and compared with the canonical EO, EO variants and other frontier metaheuristic approaches. The experimental results reveal that the developed ACEO method remarkably outperforms the canonical EO and other competitors. In addition, ACEO is implemented to solve a mobile robot path planning (MRPP) task, and compared with other typical metaheuristic techniques. The comparison indicates that ACEO beats its competitors, and the ACEO algorithm can provide high-quality feasible solutions for MRPP.

A mixed nonlinear time-fractional Rayleigh-Stokes equation

This paper investigates a nonlinear time-fractional Rayleigh-Stokes equation with mixed nonlinearity containing a power-type function, a logarithmic function and an inverse time-forcing term. Applying Lagrange's mean value theorem and the compactness of the Sobolev embeddings, we estimate the complex Lipschitz property of mixed nonlinearity. We investigate the local well-posed results (local existence, regularity estimate, continuation) of the solutions in Hilbert scales space. Moreover, the global existence theory affiliated to the finite-time blow-up is considered.

Bifurcations of traveling wave solutions for the mixed Korteweg-de Vries equation

<abstract><p>In this paper, the bifurcation theory of planar dynamical systems is employed to investigate the mixed Korteweg-de Vries (KdV) equation. Under different parameter conditions, the bifurcation curves and phase portraits of corresponding Hamiltonian system are given. Furthermore, many different types of exact traveling waves are obtained, which include hyperbolic function solution, triangular function solution, rational solution and doubly periodic solutions in terms of the Jacobian elliptic functions. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors, and could be applicable to explain diversity in qualitative features of wave phenomena.</p></abstract>

К вопросу единственности решения аналога задачи Дезина для уравнения параболо—гиперболического типа с условиями периодичности

In this paper investigated a problem with Dezin type condition for parabolic–hyperbolic mixed type equation. It is established a criterion for solution uniqueness to the problem.

A STUDY OF MIXED CONVECTIVE TERNARY HYBRID NANOFLUID FLOW OVER A STRETCHING SHEET WITH RADIATION AND TRANSPIRATION

This study explains the motion of a mixed convective ternary hybrid nanofluid flow with radiation and transpiration. Ternary hybrid nanofluid is the colloidal mixture of three district types of particles, namely, Al<sub>2</sub>O<sub>3</sub>, single-wall CNT, and graphene. The ordinary differential equations (ODEs) of the equation is yielded by converting partial differential equations (PDEs) with the help of suitable variables for analyzing the spherical, cylindrical, and platelet nanoparticles. Then the analytical solution can be conducted by using different shapes of the nanoparticles, namely, spherical, cylindrical, and platelet, to get the solution domain. Main results on the applications of suction and results of four roots are investigated. Also by using this domain fluctuation of friction, heat and mass transfer across the flow can be analyzed with various physical parameters. Increasing knowledge of these ternary hybrid nanofluids helps to improve the heat transfer rate. This is useful in dynamics of fuel and coolant in automobiles. Furthermore, the goal and novelty of the present work mainly explains the ternary hybrid nanofluid flow with three different shapes of nanoparticles in the presence of mixed convection. The given mixed convective Navier-Stokes equations given in PDE form are converted into ODE form using suitable transformation. The analytical results are expressed in terms of Prandtl number and in terms of four roots. Finally, the impact of different shapes of nanoparticles and parameters can be discussed using a graphical scenario. The main finding is that the Dufour number is greater for more values of thermal radiation. Also, transverse and tangential velocities as well as temperature and concentration profiles are greater for more values of solution domain and Darcy number.

DISCRETE WEIGHTED RESIDUAL METHODS WHICH ARE TECHNIQUES USED FOR NUMERICAL SOLUTION TO MIXED VOLTERRA-FREDHOLM FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

This study aims to solve the most general form of linear mixed Volterra-Fredholm integro-differential equations of multi-fractional order in the Caputo sense (MV-FIFDEs), which is solved by using orthogonal generalized Bernstein’s polynomial expansion with collocation and moment in the discrete weighted residual method under suitable conditions; then, the Clenshaw-Curtis formula is applied to approximate the integral terms of the equation numerically. In this work, the integro-fractional differential equations are reduced to algebraic linear equations and then to an operational matrix, and the solution of the resultant system yields the unknown Bernstein coefficients of approximation solutions. An algorithm has been created for each technique to handle the MV-FIFDEs using the described methods. Furthermore, numerical examples are presented to demonstrate and compare the techniques’ validity and applicability and comparisons with previous results. The majority of programs are performed on a computer using MATLAB v. 9.7. Keywords: Mixed Fractional Integro-Differential Equations, Caputo Derivative, Bernstein Polynomial, Collocation, Moment, Discrete Weighted Residual, Clenshaw-Curtis Formula, Volterra-Fredholm Equations DOI： https://doi.org/10.35741/issn.0258-2724.58.3.43

Can predators influence small rodent foraging activity rates immediately after wildfires?

Background Habitat complexity, predation risk, and intraspecific competition shape rodent communities and impact foraging. Wildfires remove vegetation shelter, which increases the predation risk perception and leads to changes in trophic resources availability. Consequently, prey adjust their foraging activity levels to minimise the likelihood of encounters with predators. Rodents select safe habitats and can detect predators’ scents, which allows them to reduce the predation risk when foraging. Aims To evaluate the effects of carnivore occurrence and habitat structure on rodent foraging activity rates immediately after fires using mixed models and structural equation modelling. Methods This study used 900-m linear transects to analyse environmental variables, acorn removal by rodents, and carnivore activity in three recently burnt areas. Results In areas with higher stone marten abundances, rodents removed more acorns. However, acorn removal was also higher in structurally complex habitats with greater rodent abundance. Conclusions Rodents’ foraging activity is driven by increased interspecific competition and the predation risk perception due to the simplicity of the burnt habitat. Additionally, stone martens and rodents share the same preferences for habitat complexity after fires. Implications Habitat complexity increases seed holding by rodents, which positively contributes to fire recovery and attracts predators, thereby increasing species diversity.