Second-order Partial Differential Equations Research Articles

Formation Control of Multiagent System Based on Higher Order Partial Differential Equations

We propose a method of controlling the formation in a multiagent system using partial differential equations (PDEs). In this method, the behavior of the entire multiagent system is modeled by second-order or higher order PDEs. We also propose a boundary controller that exponentially stabilizes the PDE model. By discretizing the PDE model under the proposed controller, the follower agents’ control laws can be derived. Moreover, the boundary controller corresponds to the leader agents’ control laws. The use of higher order PDEs leads to the generation of various formations that cannot be generated by using lower order PDEs. Finally, we conduct numerical simulations and experiments to validate the proposed method.

Переходные вероятности марковского процесса на отрезке, лежащем в четверти плоскости

The paper considers a quadratic birth-death Markov process. The points on a line segment located within a quarter-plane represent the states of the random process. We designate the set of vectors that have integer non-negative coordinates as our quarter plane. The process is defined by infinitesimal characteristics, or transition probability densities. These characteristics are determined by a quadratic function of the coordinates at the segment points with integer coordinates. The boundary points of the segment are absorbing; at these points, the random process stops. We investigated a critical case when process jumps are equally probable at the moment of exiting a point. We derived expressions describing transition probabilities of the Markov process as a spectral series. We used a two-dimensional exponential generating function of transition probabilities and a two-dimensional generating function of transition probabilities. The first and second systems of ordinary differential Kolmogorov equations for Markov process transition probabilities are reduced to second-order mixed type partial differential equations for a double generating function. We solve the resulting system of linear equations using separation of variables. The spectrum obtained is discrete. The eigen-functions are expressed in terms of hypergeometric functions. The particular solution constructed is a Fourier series, whose coefficients are derived by means of expo-nential expansion. We employed sums of functional series known in the theory of special functions to construct the exponential expansion required

Theoretical dynamic displacement analysis for rock bolt with surrounding rock-soil body

To analyze rock-bolt dynamic mechanics, one theoretical model has been put forward based on several assumptions. At the interface of the bolt and surrounding rock-soil body, the coupling of a linear spring and a damper is used for the bolt system. The coefficient k of the spring and the damper coefficient η are key parameters between the rock-soil body and bolt. The elastic and viscoelastic kinematic equations are established, and the displacement control equations are binary second-order partial-differential equations that can be simulated by numerical methods. One example shows the displacement curves with input waves; the displacement decays with different distributions. The curves of different times were simulated, and it was found that as the ratio value of the coefficient k of the spring and elastic modulus E increases, the displacement-decay-ratio speed increases per unit length. The same is true for the ratio values of the damper coefficient η and elastic modulus E. The theoretical model can also be used to solve the impact-force problems.

ИССЛЕДОВАНИЕ МОДЕЛИ КРОТОВОЙ НОРЫ ШВАРЦШИЛЬДА-ДАМУРА-СОЛОДУХИНА

The equations of general relativity are nonlinear second-order partial differential equations, and, as a consequence, obtaining the exact solutions is a difficult problem. One of the solutions to this problem is to obtain models with a thin self-gravitating shell. This method is used to study most of the phenomena in the theory of gravity, where the reverse effect of matter on the geometry of space-time is a key factor. Another interesting problem that can be studied using the thin shell method is the «simulation» of a black hole. Consider a system consisting of a spherically symmetric Schwarzschild black hole and a thin shell surrounding it, located at a certain fixed distance from the black hole. From the viewpoint of gravitational physics, an observer at infinity is unable to distinguish a real black hole from a wormhole with a thin shell, in which the simulation condition is satisfied. Simulation of a black hole is possible only under sufficiently stringent conditions for the parameters of the model. In particular, the shell needs to be held at a fixed radius. In the general case, such a movement of the shell is non-geodesic, and external forces are required to hold it. The radius of the shell is also a parameter that determines the possibility / impossibility of simulation. In this paper, the radius is found for the case of a Schwarzschild black hole. In particular, the paper considers a model of a wormhole obtained as a result of gluing two space-times: a Schwarzschild black hole and a Damour-Solodukhin wormhole. The latter solution differs from the Schwarzschild black hole in the parameter of the dimensionless real deviation λ and is a twice asymptotically flat regular space-time. It is shown that they can be glued along a given radius. As a result, a thin shell is formed between two glued manifolds consisting of exotic matter. Cases are considered when the thin shell is stable. It turns out that zones corresponding to the «force» constraint are more restrictive than those corresponding to the «mass» constraint.

The generalized finite difference method with third- and fourth-order approximations and treatment of ill-conditioned stars

In this paper, we solve 2D and 3D second-order partial differential equations considering the Generalized Finite Difference Method with third- and fourth-order approximations. We analyze the influence of the number of points per star and establish some values as references. We propose a new strategy to deal with ill-conditioned stars, which are frequent in higher-order approximations. This strategy uses a few points per star and presents excellent results both for detecting ill-conditioned stars and for increasing the accuracy of the numerical approximation. We apply parallel processing to the formation of stars and derivatives, and show the speedup achieved.

Discontinuous Galerkin solutions for elliptic problems on polygonal grids using arbitrary-order Bernstein-Bézier functions

In this paper, we present a symmetric interior penalty discontinuous Galerkin finite element discretization for the numerical solution of second-order elliptic partial differential equations on general polygonal meshes using a reduced-space polygonal Bernstein-Bézier functions. They form a space of interpolatory functions (vice local polynomial functions), which satisfy the Lagrange property at their interpolation points. This can enable the use of efficient DGFEM physics-based diffusion preconditioners for the first-order linear transport equation on arbitrary polygons in the computationally-challenging asymptotic diffusion limit. On a polygonal element with n vertices and polynomial degree p, there are n vertex functions, (p−1) functions per edge, and (p−1)(p−2)/2 interior functions that span the {xayb}(a+b)≤p space of bivariate monomials. Numerical experiments highlighting the performance of the proposed method are presented through uniform and adaptive h, p, and hp refinement strategies. Appropriate error rates are observed including hp adaptivity yielding exponential convergence in the presence of singularities in the solution.

Adaptive estimator for a parabolic linear SPDE with a small noise

We deal with parametric estimation for a parabolic linear second-order stochastic partial differential equation (SPDE) with a small dispersion parameter based on high-frequency data which are observed in time and space. Using the thinned data with respect to space obtained from the high-frequency data, the minimum contrast estimators of two coefficient parameters of the SPDE are proposed. With these estimators and the thinned data with respect to time obtained from the high-frequency data, we construct an approximation of the coordinate process of the SPDE. Using the approximate coordinate process, we obtain the adaptive estimator of a coefficient parameter of the SPDE. Moreover, we give simulation results of the proposed estimators of the SPDE.

A mixed finite element method with reduced symmetry for the standard model in linear viscoelasticity

We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the main variable to be found. The resulting variational formulation is shown to be well-posed, and a class of $$\text {H}(\text {div})$$ -conforming semi-discrete schemes is proved to be convergent. Then, we use the Newmark trapezoidal rule to obtain an associated fully discrete scheme, whose main convergence results are also established. Finally, numerical examples illustrating the performance of the method are reported.

Influence of lower-order derivatives on solvability Dirichlet problem for elliptic systems

The article considers the Dirichlet problem for a multidimensional elliptic system containing lower-order derivatives. At the present time elliptic systems of second-order partial differential equations with two independent are well known, what cannot be said about multidimensional elliptic systems. The problem of homotopy classification of such systems has not yet been solved. For elliptic systems phenomena are observed in the nature of the solvability of classical boundary systems, which do not occur in the case of one equation. Among such phenomena, the influence of the loss of smoothness and the influence of lower-order derivatives on the solvability of boundary value problems should be noted. The paper proposes a solution to the Dirichlet problem for a multidimensional elliptic system with lower-order terms of a special form in a half-space. Applying the Fourier transform, the solvability problem is reduced to study of a second-order partial differential equation containing lower-order derivatives. Violations of the violation of the Noetherian property of the Dirichlet problem are considered. The result is formulated as a theorem.

Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

This study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order Partial Differential Equations (PDEs) with the aid of the method of lines. The power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derivedschemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature.

The Partial Transformational Decomposition Method for a Hybrid Analytical/Numerical Solution of the 3D Gas-Flow Problem in a Hydraulically Fractured Ultralow-Permeability Reservoir

SummaryThe analysis of gas production from fractured ultralow-permeability (ULP) reservoirs is most often accomplished using numerical simulation, which requires large 3D grids, many inputs, and typically long execution times. We propose a new hybrid analytical/numerical method that reduces the 3D equation of gas flow into either a simple ordinary-differential equation (ODE) in time or a 1D partial-differential equation (PDE) in space and time without compromising the strong nonlinearity of the gas-flow relation, thus vastly decreasing the size of the simulation problem and the execution time.We first expand the concept of pseudopressure of Al-Hussainy et al. (1966) to account for the pressure dependence of permeability and Klinkenberg effects, and we also expand the corresponding gas-flow equation to account for Langmuir sorption. In the proposed hybrid partial transformational decomposition method (TDM) (PTDM), successive finite cosine transforms (FCTs) are applied to the expanded, pseudopressure-based 3D diffusivity equation of gas flow, leading to the elimination of the corresponding physical dimensions. For production under a constant- or time-variable rate (q) regime, three levels of FCTs yield a first-order ODE in time. For production under a constant- or time-variable pressure (pwf) regime, two levels of FCTs lead to a 1D second-order PDE in space and time. The fully implicit numerical solutions for the FCT-based equations in the multitransformed spaces are inverted, providing solutions that are analytical in 2D or 3D and account for the nonlinearity of gas flow.The PTDM solution was coded in a FORTRAN95 program that used the Laplace-transform (LT) analytical solution for the q-problem and a finite-difference method for the pwf problem in their respective multitransformed spaces. Using a 3D stencil (the minimum repeatable element in the horizontal well and hydraulically fractured system), solutions over an extended production time and a substantial pressure drop were obtained for a range of isotropic and anisotropic matrix and fracture properties, constant and time-variable Q and pwf production schemes, combinations of stimulated-reservoir-volume (SRV) and non-SRV subdomains, sorbing and nonsorbing gases of different compositions and at different temperatures, Klinkenberg effects, and the dependence of matrix permeability on porosity. The limits of applicability of PTDM were also explored. The results were compared with the numerical solutions from a widely used, fully implicit 3D simulator that involved a finely discretized (high-definition) 3D domain involving 220,000 elements and show that the PTDM solutions can provide accurate results for long times for large well drawdowns even under challenging conditions.Of the two versions of PTDM, the PTD-1D was by far the better option and its solutions were shown to be in very good agreement with the full numerical solutions, while requiring a fraction of the memory and orders-of-magnitude lower execution times because these solutions require discretization of only the time domain and a single axis (instead of three). The PTD-0D method was slower than PTD-1D (but still much faster than the numerical solution), and although its solutions were accurate for t < 6 months, these solutions deteriorated beyond that point.The PTDM is an entirely new approach to the analysis of gas flow in hydraulically fractured ULP reservoirs. The PTDM solutions preserve the strong nonlinearity of the gas-flow equation and are analytical in 2D or 3D. This being a semianalytical approach, it needs very limited input data and requires computer storage and computational times that are orders-of-magnitude smaller than those in conventional (numerical) simulators because its discretization is limited to time and (possibly) a single spatial dimension.

Optimal Transport of Information

We study the general problem of Bayesian persuasion (optimal information design) with continuous actions and continuous state space in arbitrary dimensions. First, we show that with a finite signal space, the optimal information design is always given by a partition. Second, we take the limit of an infinite signal space and characterize the solution in terms of a Monge-Kantorovich optimal transport problem with an endogenous information transport cost. We use our novel approach to: 1. Derive necessary and sufficient conditions for optimality based on Bregman divergences for non-convex functions. 2. Compute exact bounds for the Hausdorff dimension of the support of an optimal policy. 3. Derive a non-linear, second-order partial differential equation whose solutions correspond to regular optimal policies. We illustrate the power of our approach by providing explicit solutions to several non-linear, multidimensional Bayesian persuasion problems.

An accurate and novel numerical simulation with convergence analysis for nonlinear partial differential equations of Burgers–Fisher type arising in applied sciences

Abstract In this study, the second-order nonlinear partial differential equations of Burgers–Fisher type are considered under a unique formulation by introducing a novel highly accurate numerical method based on the Nörlund rational polynomial and matrix-collocation computational system. The method aims to obtain a sustainable approach since it contains the rational structure of the Nörlund polynomial. A unique computer program module, which involves very few routines, is constructed to discuss the precision and efficiency of the method and these few steps are described via an algorithm. A residual function is employed in both the error and convergence analyses with mean value theorem for double integrals. The considered equations in the numerical tests stand for model phenomena arising widely in applied sciences. Graphical and numerical comparisons provide a clear observation about the consistency of the method. All results prove that the method is highly accurate, eligible, and provides the ultimate operation for aforementioned problems.

Parametric design of purely compressed shells

Within the frame of parametric design, in the present work we focus on a very special objective, namely parametrically generating families of purely compressed shells. A similar task can be pursued by adopting for the equilibrium analysis the so-called Thrust Network Analysis for which the shell structure is condensed into a network of bars. Here instead, we adopt a continuum approach, namely the so-called Membrane Equilibrium Analysis. With this continuum approach, a purely compressed membrane equilibrium solution is searched by solving a scalar second-order partial differential equation representing the transverse equilibrium equation of the membrane. The shell is compressed if the membrane surface is contained within the volume of the shell and if the stress potential is concave. For a given shell, the main difficulty is represented by the fulfillment of the concavity constraint for the stress potential. In the present study, this difficulty is overcome by assigning families of convenient concave stress potentials and considering the shape as the unknown. By considering stress potentials or boundary data controlled by a few parameters, such variable parameters can be manipulated in order to alter the end result. Other methods tackling the stress function with the help of a computer, exist in the literature, but the main contribution of the present paper is the shape analysis of compression-only shells with the help of finite element apparatus. A few illustrative examples are presented to demonstrate the method.

RETRACTED ARTICLE: Modeling study on influence of surface pore water on slope stability of mountain tunnel

In order to solve the problems of slope strength reduction and stability risks caused by excessive surface water in mountain tunnels, the influence of surface pore water on slope stability is deeply studied. This article first analyzes the seepage field of unsaturated soil. Taking the transient seepage field as the object, the distribution of the soil seepage field during the two-stage precipitation process is studied. Secondly, the existing dimensionless second-order partial differential equation is solved by Laplace transform to calculate the distribution of transient seepage field. A finite element model of rock and soil seepage field under different rainfall intensities was established, and the influence of pore water content on slope stability was quantitatively analyzed. The results show that as the anti-sliding force decreases, the pore water content increases, the pressure increases, the suction force of the matrix decreases, the shear strength decreases, and the slope safety factor decreases, which leads to the plastic deformation of the slope.

Finite state mean field games with Wright–Fisher common noise

We force uniqueness in finite state mean field games by adding a Wright–Fisher common noise. We achieve this by analyzing the master equation of this game, which is a degenerate parabolic second-order partial differential equation set on the simplex whose characteristics solve the stochastic forward-backward system associated with the mean field game; see Cardaliaguet et al. [10]. We show that this equation, which is a non-linear version of the Kimura type equation studied in Epstein and Mazzeo [28], has a unique smooth solution whenever the normal component of the drift at the boundary is strong enough. Among others, this requires a priori estimates of Hölder type for the corresponding Kimura operator when the drift therein is merely continuous.

Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type

This article is mainly concerned with the existence and the forms of entire solutions for several systems of the second-order partial differential difference equations of Fermat type α ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 1 z 1 , z 2 / ∂ z 2 2 n 1 + f 2 z 1 + c 1 , z 2 + c 2 m 1 = 1 α ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 2 z 1 , z 2 / ∂ z 2 2 n 2 + f 1 z 1 + c 1 , z 2 + c 2 m 2 = 1 and ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 2 + f 2 z 1 + c 1 , z 2 + c 2 2 = 1 ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 2 + f 1 z 1 + c 1 , z 2 + c 2 2 = 1 . Our results about the existence and the forms of solutions for these systems generalize the previous theorems given by Xu and Cao, Gao, Liu, and Yang. In addition, we give some examples to explain the existence of solutions of this system in each case.

Rotational modes of Poincaré Earth models

ABSTRACT We study the following rotational modes of Poincaré Earth models: the tilt-over mode (TOM), the spin-over mode (SOM) and free core nutation (FCN), using first a simple Earth model with a homogeneous and incompressible liquid core (LC) and a rigid mantle (MT). We obtain analytical solutions for the periods of these modes as well as that of the Chandler wobble (CW). We show analytically the distinction between the TOM and the SOM and that the FCN is indeed the same mode as the SOM of a wobbling Earth. The reduced pressure, in terms of which the vector momentum equation is known to reduce to a scalar second-order partial differential equation called the Poincaré equation, is used as the independent variable. Analytical solutions are then found for the displacement eigenfunctions in a meridional plane of the liquid core for the aforementioned modes. We next consider a three-layer Earth model similar to above which also includes a rigid inner core (IC). We first show that analytical solutions exist for the period and eigenfunctions of the CW if the IC is locked to the MT, i.e. they have the same wobbling motion. We show that this is significant as it shows that the CW manifests itself for a Poincaré (incompressible and inviscid LC) wobbling Earth model. We further allow for the inner core to wobble independently and compute numerically the periods and displacement eigenfunctions of the TOM, SOM and FCN, as well as those for still another rotational mode, the inner-core wobble (ICW). Next we show that the presence of the characteristic surfaces intercepted by the inner-core, when computing the period and eigenfunctions of the free inner-core nutation (FICN), may be the reason for the slow (or lack of the) convergence of this mode. Finally, we show that even though the wobbling motion of the mantle is ignored when solving for the frequencies of the ICW and the FICN when Sasao's approximation is used, the analytical solutions for both these modes yield periods nearly identical to those in the literature for a similar Earth model with mantle allowed to wobble as well. We infer that the Sasao's approximation, or the severe truncation of the series solution of the field variables, the pressure, the gravitational potential and the components of the displacement vector, may not be adequate to accurately describe the motion in the liquid core during the excitation of the FICN.

A new (3 + 1)-dimensional Painlevé-integrable Sakovich equation: multiple soliton solutions

PurposeThis paper aims to develop a new (3 + 1)-dimensional Painlevée-integrable extended Sakovich equation. This paper formally derives multiple soliton solutions for this developed model.Design/methodology/approachThis paper uses the simplified Hirota’s method for deriving multiple soliton solutions.FindingsThis paper finds that the developed (3 + 1)-dimensional Sakovich model exhibits complete integrability in analogy with the standard Sakovich equation.Research limitations/implicationsThis paper addresses the integrability features of this model via using the Painlevée analysis. This paper reports multiple soliton solutions for this equation by using the simplified Hirota’s method.Practical implicationsThe study reports three non-linear terms added to the standard Sakovich equation.Social implicationsThe study presents useful algorithms for constructing new integrable equations and for handling these equations.Originality/valueThe paper reports a new Painlevée-integrable extended Sakovich equation, which belongs to second-order partial differential equations. The constructed model does not contain any dispersion term such as uxxx.

Degree for weakly upper semicontinuous perturbations of quasi-m-accretive operators.

In the paper, we provide the construction of a coincidence degree being a homotopy invariant detecting the existence of solutions of equations or inclusions of the form Ax ∈ F(x), x ∈ U, where [Formula: see text] is an m-accretive operator in a Banach space E, [Formula: see text] is a weakly upper semicontinuous set-valued map constrained to an open subset U of a closed set K ⊂ E. Two different approaches are presented. The theory is applied to show the existence of non-trivial positive solutions of some nonlinear second-order partial differential equations with discontinuities. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.