Partial Differential Research Articles

Trypanosomosis and Transhumance: Contributions to Contemporary Conflicts Between Farmers and Herdsmen Along the Tsetse Fly Belts: Mathematical Modeling and Systematic Field Analysis Approach.

Conflicts within the tsetse fly belt revealed a strong correlation between the dynamics of bovine trypanosomosis and the insurgency involving farmers and herders in Nigeria and parts of West Africa. This study examined the history, causes and influence of farmers-herdsmen conflicts on banditry, terrorism and food security as it relates to the epidemiology of African animal trypanosomosis (AAT). A combination of literature database searches, semi-structured questionnaires, and mathematical modeling was employed. The study found that transhumance contributes significantly to conflicts between farmers and herdsmen. An average of 6.46 persons per attack were reported between 2005 and 2021. Only 8.4 (95 CI: 5.0 12.9) of farmers and 18.2 (95 CI: 12.4 25.4) of herdsmen have engaged in conflict resolution efforts. The study shows that both conflict and the spread of trypanosomosis can be effectively controlled when , ensuring that the sub-population remains in the basin of attraction of the trypanosomosis-conflict-free equilibrium ( ). The partial derivative of the basic reproduction number, , with respect to improved conflict resolution, suggests that halting transhumance can prevent a portion of the cattle recruitment rate ( ) from becoming infected with AAT. Climate change exacerbates these issues, leading to settlement and resettlement strategies within the fly belt regions. The model indicates that the basic reproduction number can only be reduced to less than one ( ) to become globally asymptotically stable if there is effective conflict resolution involving both farmers and herders. The study advocates for the establishment of ranching in tsetse-free zones with adequate social amenities, improved marketing strategies for animals and animal products led by government agencies through public-private partnerships, the banning of open grazing, and strict enforcement of policies against violators.

Ligand and Residue Free Energy Perturbations Solve the Dual Binding Mode Proposal for an A2BAR Partial Agonist.

Adenosine receptors, particularly A2BAR, are gaining attention for their role in pathological conditions such as cancer immunotherapy, prompting the exploration for promising therapeutic applications. Despite numerous selective A2BAR antagonists, the lack of selective full agonists makes the partial agonist BAY60-6583 one of the most interesting activators of this receptor. Recent cryo-EM structures have univocally revealed the binding mode of nonselective ribosidic agonists such as adenosine and its derivative NECA to A2BAR; however, two independent structures with BAY60-6583 show alternative binding orientations, raising the question of which is the physiologically relevant binding mode. In situations such as this, computational simulations that accurately predict shifts in binding free energy can complement experimental structures. Our study combines QligFEP and QresFEP protocols to directly compare the binding affinity of BAY60-6583 between alternative binding modes as well as providing a direct comparison of in silico mutagenesis studies on each pose with experimental mutational effects. Both methods converge on the experimentally determined binding mode that better explains both the existing SAR and mutagenesis data for this ligand. Our results allow the elucidation of the experimental binding orientation that should be considered as a basis for designing partial agonist derivatives with improved affinity and selectivity and underscore the value of free energy perturbation methods in aiding structure-based drug design.

DERIVATIVE-BASED SHAPLEY VALUE FOR GLOBAL SENSITIVITY ANALYSIS AND MACHINE LEARNING EXPLAINABILITY

We introduce a new Shapley value approach for global sensitivity analysis and machine learning explainability. The method is based on the first-order partial derivatives of the underlying function. The computational complexity of the method is linear in dimension (number of features), as opposed to the exponential complexity of other Shapley value approaches in the literature. Examples from global sensitivity analysis and machine learning are used to compare the method numerically with activity scores, SHAP, and KernelSHAP.

Variant of the Mathematical Theory of Non-Thin Multilayer Nonlinearly Elastic Plates of Symmetric Structures

A new version of the mathematical theory of non-thin multilayer nonlinearly elastic Kauderer plates of symmetric structures has been constructed. The components of the stress-strain state (SSS) and boundary conditions on the side surface of the plate are considered functions of three spatial coordinates. The variant is based on the development of components of the SSS of the plate along the transverse coordinate with the help of combinations of Legendre polynomials within each layer. The three-dimensional physically nonlinear boundary value problem for a multilayer non-thin plate is reduced to a two-dimensional one using the three-dimensional equations of the theory of elasticity and Reissner's variational principle. The main dependencies are obtained. Derived systems of high-order differential equations of equilibrium with partial derivatives and boundary conditions containing nonlinear terms from the SSS. The conjugation conditions at the boundary of adjacent layers (rigid connection) are exactly fulfilled for displacements, transverse tangential and normal stresses. The boundary conditions for stresses on the horizontal faces of the plate are also exactly fulfilled. The system of high-order differential equations of equilibrium is transformed into two systems that independently describe skew-symmetric and symmetric deformation relative to the median plane of the plate. In turn, each of these systems is transformed into differential equations that describe vortex boundary effects and equations that describe an internal SSS with a potential boundary layer-type boundary effect. The solution of systems of high-order equations by the developed method is reduced to the solution of second-order differential equations (Poisson and Helmholtz equations). General solutions of the system of differential equilibrium equations are obtained. The validity and accuracy of the variant of the mathematical theory is confirmed by comparisons with exact solutions of bending problems for linearly elastic plates with different mechanical and geometrical parameters. The newly constructed variant of the theory and the developed method provide a real opportunity to solve boundary problems in a spatial formulation with high accuracy for multilayer platens under various loads and boundary conditions on the lateral surface.

THE RESEARCH OF AN IMPACT IN STRENGTH CHARACTERISTICS OF OIL-CONTAMINATED SANDY SOILS

A review of existing studies by domestic and foreign authors on changes in the angle of internal friction of sandy soils contaminated with oil products is provided. The main experimentally confirmed results of studies of the angle of internal friction of sandy soils interacting with oil products are outlined. Laboratory tests were conducted to determine the strength characteristics of sandy soils contaminated with oil products using the in-plane shear method. Coarse, medium and fine sands contaminated with A-95 gasoline, diesel fuel, and heavy oil at a concentration of 0-12% by dry soil weight were used as test samples. A methodology for conducting laboratory tests using the in-plane shear method for sands contaminated with oil products is described and tested. The degree of influence of viscosity and oil product concentration factors on the value of the angle of internal friction of sands of different sizes was determined and mathematically substantiated using regression multifactorial analysis. The sensitivity analysis of the adopted regression equation with six degrees of freedom is performed. The scope of application of the proposed function is designated. It is shown that the use of polynomials of the second kind in linear regressions leads to a significant complication of the model associated with the impossibility of explicitly identifying individual influence coefficients without additional calculation of partial derivatives to compile the function gradient with its subsequent analysis. A decrease in the angle of internal friction of coarse, medium and fine sandy soils regardless of the type of oil product and its concentration has been laboratory confirmed. A graph of the dependence of the angle of internal friction of sands of different sizes on the concentration of oil products is presented. It has been statistically substantiated that the greatest influence on the angle of internal friction of sandy soils contaminated with oil products is exerted by concentration, to a lesser extent - the type (viscosity) of oil products

Compressed Structures for Partial Derivative Automata Constructions

The partial derivative automaton ([Formula: see text]) is an elegant simulation of a regular expression. Although it is, in general, smaller than the position automaton ([Formula: see text]), the algorithms that build [Formula: see text] in quadratic worst-case time first compute [Formula: see text]. Asymptotically, and on average for the uniform distribution, the size of [Formula: see text] is half the size of [Formula: see text], being both linear on the size of the expression. We address the construction of [Formula: see text] efficiently, on average, avoiding the computation of [Formula: see text]. The expression and the set of its partial derivatives are represented by a directed acyclic graph with shared common subexpressions. We develop an algorithm for building [Formula: see text]s from expressions in strong star normal form of size [Formula: see text] that runs, on average, in time that asymptotically behaves as [Formula: see text], and space as [Formula: see text]. Empirical results corroborate its good practical performance.

On the use of direct implicit integration with numerical Jacobian matrix in helicopter rotor dynamics

In recent years, geometrically exact beam theory is widely used in rotor structural dynamics to model composite blade undergoing large deflections. The corresponding differential equations based on this theory show strong nonlinearity and the aeroelastic terms are normally written in implicit format. It brings most difficulties to implicit integration methods since the analytical Jacobian matrix is hard to derive. In this paper, a trapezoidal integration utilizing numerical Jacobian matrix is introduced to solve the dynamic equations of rotor blade. Central difference method with Romberg extrapolation scheme is adopted to numerically calculate the Jacobian. Accuracy of the methods is evaluated on dynamic problems from one cantilevered beam and the other fully articulated rotor blade. Numerical stability and efficiency are also investigated by changing update policies of Jacobian matrix and mesh configurations of typical rotor blade. Numerical results show that both extrapolation scheme for partial derivatives and the integration method with numerical Jacobian perform well on predicting natural frequencies and transient responses in rotor dynamics.

The field emission characteristics under transient electric field

This article concentrates on the field emission (FE) characteristics under the pulsed transient electric field. Experimental measurements are carried out by applying direct current (DC) voltage, millisecond pulse voltage, and microsecond pulse voltage. Additionally, 304 stainless steel, oxygen-free copper and titanium electrodes are utilized to verify the consistency. Compared with the case under DC electric field, three distinctive FE characteristics are observed under pulsed transient electric field: the regular emission, the intense emission, and the annihilation phenomenon. First, the emission starting point implies one strong correlation with the second partial derivative of transient electric field strength with respect to time. Second, the emission current under pulsed electric field is much higher than that under the DC electric field. Moreover, the FE current is deeply associated with the gradient of the electric field during the rising front. Third, the FE current is extinguished though there is still high transient electric field. The mechanism of the three characteristics is still unclear and should be the subject of further investigation.

MATHEMATICAL MODELING OF THE DYNAMICS OF THE AEROELASTIC "PIPELINE – PRESSURE SENSOR" SYSTEM

This paper proposes mathematical models of mechanical systems "pipeline - pressure sensor" designed to control the pressure of the working medium in the combustion chambers of engines. In such systems, to mitigate the impact of vibration accelerations and high temperatures, the sensor is connected to the engine using a pipeline and is located at some distance from it. The movement of the working medium is described by linear models of fluid and gas mechanics. To describe the dynamics of an elastic sensitive element, linear models of the mechanics of a solid deformable body are used. Based on linear differential equations with partial derivatives, mathematical formulations of problems are proposed that correspond to three-dimensional models of pressure measurement systems in gas-liquid media for some pipeline cross-sectional shapes, namely, for a pipeline with a rectangular cross-section, with a section in the form of a sector and in the form of a ring. By introducing integral characteristics, the solution of problems is reduced to studying one-dimensional models. Equations have been obtained that make it possible to determine the pressure of the working medium in the combustion chamber at each moment of time by the value of deformation of the sensitive element of the sensor. Analytical and numerical-analytical methods for solving the corresponding initial-boundary value problems for systems of differential equations are proposed. In the analytical approach, the solution of the problem is reduced to solving a differential equation with a deviating argument. The numerical-analytical study of the problem is based on the application of the Galerkin method. Also, a numerical experiment was carried out and examples of calculating the deformation of the sensitive element of the sensor in the case of rigid fastening when specifying specific values of the mechanical parameters of the system are presented, also during setting the law of change in the excess pressure of the working medium in the engine.

BOUNDARY PROBLEM FOR THE LAPLACE EQUATION WITH MIXED BOUNDARY CONDITIONS IN A SEMIBAND

Theorems on the existence and uniqueness of the solution to the Laplace equation with mixed boundary conditions in a semiband have been proven in the work. Additionally, integral representations for the partial derivatives of the solution have been obtained.

On a mixed partial Caputo derivative and its applications to a hyperbolic partial fractional differential equation

AbstractWe propose an alternative definition of a mixed partial derivative in the Caputo sense for functions of two variables defined on the rectangle $$P=[0,a]\times [0,b]$$ P = [ 0 , a ] × [ 0 , b ] ($$a>0, b>0$$ a > 0 , b > 0 ). We give an integral representation of functions possessing such a derivative. Moreover, we study the existence and uniqueness of a solution, as well as the Ulam–Hyers type stability of a fractional counterpart of a nonlinear continuous Goursat-Darboux system described by the introduced Caputo derivative. This paper is a continuation of our paper [R. Kamocki, C. Obczyński, On the single partial Caputo derivatives for functions of two variables, Periodica Mathematica Hungarica 87(2), (2023), 324–339].

Unifying the thermodynamic and electrostatic view on the pyroelectric effect

Pyroelectricity is a first rank tensor (vector) property that connects electric displacement to temperature. Like all vector properties, it is allowed only in certain point group symmetries. In related literature, a number of formulations exist that apparently treat various contributions to the pyroelectric coefficient but without strictly considering the crystal symmetry. We revisit the formulation of the pyroelectric coefficient in the presence of external fields and find that a consistent treatment of the pyroelectric coefficient, allowing one to arrive at a single convergent formula starting from either thermodynamic or electrostatic arguments, may not be straightforward. Motivated by this outcome, we develop an approach allowing both electrostatic and thermodynamic arguments for pyroelectricity to converge to one single expression with mathematical consistency of the partial derivatives. Albeit not very significant at first sight, the approach and the manner in which fields are introduced prove vital in evaluating the field contributions to the pyroelectric effect and their deconvolution in experiments. Importance of the crystal point groups is also highlighted in the context of vector properties, specifically pyroelectric coefficient. Finally, the so-called induced pyroelectricity and a correct mathematical expansion of the polarization is discussed to clarify the contributions of the pyroelectric effect.

CONDITIONS OF PARABOLIC AND HYPERBOLIC TYPES OF NONPARAMETRIC SURFACE

A nonparametric two-dimensional surface is considered, i.e. the graph of some smooth function 𝑓 = 𝑓(𝑥, 𝑦) defined over the entire plane. The surface type is defined in terms of the two-dimensional capacity of a compact set on the surface as follows. If the capacity of any non-degenerate continuum on the surface is zero, then the surface has a parabolic type, and if the capacity of any non-degenerate continuum on the surface is positive, then the surface has a hyperbolic type. The paper establishes integral conditions for the function 𝑓 defining the surface, under which the surface has one or another type. These conditions are expressed as conditions of convergence or divergence of the corresponding integrals and characterize the degree of growth (for ρ → +∞) of the partial derivatives |𝑓′ ρ | and |𝑓′ φ|, where ρ and φ are the polar coordinates of a point (𝑥, 𝑦) of the plane over which the surface lies. It turns out that if the degree of growth of the function |𝑓′ φ | is small, or more precisely, less than the growth of a linear function of ρ, then, regardless of the growth rate of |𝑓′ ρ |, the surface is of parabolic type. But if (for ρ → +∞) the growth of the function |𝑓′ φ| exceeds the linear growth of ρ, and at the same time exceeds the growth of |𝑓′ ρ|, then the surface is of hyperbolic type. Using the obtained conditions of the hyperbolic type of surface, the construction of an example of a nonparametric surface of hyperbolic type is described. This example complements the known examples of this type constructed by various famous mathematicians back in the late 50s of the last century. But the justifications of these examples, unlike the example proposed in this paper, do not rely (explicitly) on any general condition of the hyperbolic type of surface.

Optimal local truncation error method on unfitted Cartesian meshes for solution of 3-D wave and heat equations for heterogeneous materials

In the paper we develop the optimal local truncation error method (OLTEM) with the non-diagonal and diagonal mass matrices on unfitted Cartesian meshes for the 3-D time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. 27-point stencils that are similar to those for linear finite elements are used with OLTEM. There are no unknowns for OLTEM on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations, includes the use of the interface conditions and the entire PDE in derivations, and yields the optimal fourth order of accuracy of OLTEM with the non-diagonal mass matrix. This is two order higher than for linear finite elements with similar 27-point stencils. OLTEM with the diagonal mass matrix includes the rigorous calculation of the diagonal mass matrix and provides the optimal second order of accuracy. The 3-D numerical results for heterogeneous materials with irregular interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than high-order (up to the 7-th order - the highest order in the commercial code COMSOL) finite elements with much wider stencils. Compared to linear finite elements with similar 27-point stencils, at an accuracy of 0.1% OLTEM decreases the number of degrees of freedom by 550–7300 times. This leads to a huge reduction in computation time.A new 3-D post-processing procedure has been developed with OLTEM for the calculation of the spatial derivatives of time-dependent numerical solutions. The spatial derivatives for each grid point are calculated with the help of one compact 27-point stencil (the same as that in basic computations). In contrast to known post-processing procedures, the new approach includes also the use of the time derivatives of the function, the interface conditions and the original PDE. The spatial derivatives of the OLTEM solutions calculated with the new post-processing procedure are much more accurate compared to those obtained by high-order (up to the 7-th order) finite elements with much wider stencils. At an accuracy of 0.1% for the spatial derivatives, OLTEM decreases the number of degrees of freedom by 1.9⋅106–6⋅109 times compared to linear finite elements. The new post-processing procedure can be equally applied to the calculation of the partial derivatives obtained by other numerical methods as well as to the numerical results for other PDEs.Due to the huge reduction in the computation time compared to existing methods and the use of trivial unfitted Cartesian meshes that are independent of irregular geometry, the proposed technique does not require remeshing for the shape change of irregular geometry and it will be effective for many design and optimization problems as well as for multiscale problems without the scale separation.

Finite Element Method applied to the Fractional Partial Derivative Equation of Anomalous Diffusion

Diffusion, a fundamental phenomenon in nature, operates on both microscopic and macroscopic scales. At the microscopic level, it manifests itself as a stochastic process, while at the macroscopic level it represents a uniform drift toward equilibrium. The Fokker-Planck equation is widely used to model diffusion phenomena in various disciplines, including physics, chemistry, biology, and others. The classical Fick’s law cannot correctly represent the movement of species through inhomogeneous materials, such as porous media. This phenomenon is known as non-Fickian or anomalous diffusion and presents behavior that is difficult to analyze and require computationally intensive simulations. To address this, recent studies have explored modeling anomalous diffusion using fractional derivatives in time or space. Fractional calculus, a branch of classical calculus, provides a framework for handling integrals and derivatives of arbitrary order. Its application has proven to be particularly effective in systems that exhibit hysteresis, allowing the computation of associated memory effects. One example is the modeling of anomalous diffusion in polymeric coatings used to protect flexible pipelines in subsea oil exploration. Under such adverse conditions, extreme depths and temperatures, the properties of the polymer change over time, affecting the ingress of corrosive ions into the metal structure. It left unchecked, this threatens operational safety and environmental integrity. In this work, the Finite Element Method (FEM) is applied to anomalous diffusion described by Fractional Partial Derivative Equations (FPDEs), seeking to predict these intricate transport dynamics.

Information hidden in the state change of a gas obtained from the theoretical integrations of the partial derivatives

Information hidden in the state change of a gas obtained from the theoretical integrations of the partial derivatives

Topological, Differential Geometry Methods and Modified Variational Approach for Calculation of the Propagation Time of a Signal, Emitted by a GPS-Satellite and Depending on the Full Set of 6 Kepler Parameters

Abstract In preceding publications a mathematical approach has been developed for calculation of the propagation time of a signal, emitted by a moving along an elliptical orbit satellite and accounting also for the General Relativity Theory (GRT) Effects. So far, the formalism has been restricted to one dynamical parameter (the true anomaly or the eccentric anomaly angle). In this paper the aim is to extend the formalism to the case, when also the other five Kepler parameters will be changing and thus, the following important problem can be stated: if two satellites move on two space-distributed orbits and they exchange signals, how can the propagation time be calculated? This paper requires the implementation of differential geometry and topological methods. In this approach, the action functional for the propagation time is represented in the form of a quadratic functional in the differentials of the Kepler elements, consequently the problem is related to the first and second quadratic forms from differential geometry. Such an approach has a clear advantage, because if the functional is written in terms of Cartesian coordinates X, Y, Z, the extremum value after the application of the variational principle is shown to be the straight line - a result, known from differential geometry, but not applicable to the current problem of signal exchange between satellites on different orbits. So the known mapping from celestial mechanics is used, when by means of a transformation the 6 Kepler parameters are mapped into the cartesian coordinates X, Y, Z. This is in fact a submersion of a manifold of 6 parameters into a manifold of 3 parameters. If a variational approach is applied with respect to a differential form in terms of the differentials of the Kepler parameters, the second variation will be different from zero and the Stokes theorem can be applied, provided that the second partial derivatives of the Cartesian coordinates with respect to the Kepler parameters are assumed to be different from zero. ‘From topology this requirement is equivalent to the existence of the s.c. Morse functions (non-degenerate at the critical points). In the given case it has been shown that Morse function cannot exist with respect to each one of the Kepler parameters- Morse function cannot be defined with respect to the omega angle.

Simulation of Scalar Wave Propagation with High-Order Temporal and Spatial Accuracy by a New Multi-Axial Staggered-Grid Finite-Difference Scheme

ABSTRACT Staggered-grid finite-difference (SFD) stencils are extensively applied for scalar wavefield simulations and inversions in seismology because of their easy implementation and effectiveness of propagating the wave in heterogeneous media. The conventional SFD (CSFD) stencil adopts second-order temporal and high-order spatial finite-difference operators to approximate the partial derivatives inside the wave equation. The spatial SFD operator only adopts grid points along one orthogonally axial direction to approximate the spatial partial derivative along that direction. Therefore, increasing the number of grid points along the axis will not improve the temporal accuracy. To simultaneously enhance the temporal and spatial accuracy, we propose a new multi-axial SFD (MASFD) stencil, which consists of grid points along three directions for each partial derivative in space. The MASFD weightings (coefficients) are derived by preserving the dispersion relation of the scalar wave in the frequency–wavenumber domain. We prove that increasing the number of the grid points of the new stencil can simultaneously reach high-order accuracy in time and space. The performance of the new MASFD scheme is compared with the CSFD schemes by quantitative dispersion analyses, stability analyses, and numerical examples. Our comprehensive comparisons demonstrate that the MASFD scheme can be more accurate than the CSFD ones because of improved temporal accuracy. Under comparable accuracy, the MASFD scheme can be more efficient than the CSFD ones because the MASFD scheme can adopt larger time steps to perform stable wave extrapolation.

Normalization of hot deformation parameters for activation energy analysis in metallic alloys

Abstract Depending on melting point of a given alloy, the temperature range for
hot deformation of alloys varies widely. Consequently, the apparent activation energy
of hot deformation, Q, is greatly influenced by the temperature range as well as strain
rate applied during hot deformation. The purpose of this investigation is to normalize
hot deformation parameters – temperature and strain rate – to study their effects on
activation energy. Seventy published hot-deformation data sets were used to investigate
the effects of normalized hot-deformation variables on Arrhenius-type constitutive
model parameters. Flow stress, strain rate, and temperature from sample datasets
were normalized to the range [0, 1] and fitted with an Arrhenius-type model. The
results showed that, for normalized parameters, the activation energy has exponential
relationships with a partial derivative of flow stress with strain rate and temperature.

A Spectral Element Solution of the Poisson Equation with Shifted Boundary Polynomial Corrections: Influence of the Surrogate to True Boundary Mapping and an Asymptotically Preserving Robin Formulation

We present a new high-order spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order hp-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series, which traditionally is used within the shifted boundary method. Here, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small cut cells. Dirichlet, Neumann, and Robin boundary conditions are enforced weakly through a generalized: (i) Nitsche’s method and (ii) Aubin’s method. A consistent asymptotic preserving formulation of the embedded Robin formulations is presented. Several experiments and analyses of the numerical properties of the various weak forms are showcased. We include convergence studies under polynomial increase of the basis functions, p, mesh refinement, h, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. With this, we assess the influence of errors across variational forms, polynomial order, mesh size, and mappings between the true and surrogate boundaries.