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In this article, the third boundary value problem for an equation with the third order multiple characteristics in three dimensional space is studied. Uniqueness of the solution of the given problem is proved by the method of energy integral. Existence of the solution is proved by means of the method of variables separation. The solution is constructed in the exact infinite series form and an opportunity of term-by-term differentiation is justified. It is important in smooth convergence of the series, that "small denominator" is different from zero. [ABSTRACT FROM AUTHOR]

A new proof of Hartman and Winter's theorem (Am. J. Math. 74(4): 834–864, 1952) is presented by using fixed point theory. An illustrative example to this theorem is also given. [ABSTRACT FROM AUTHOR]

Nowadays, modern gas supply systems are complex. They consist of gas distribution stations; high-, medium-, and low-pressure gas networks; gas installations; and control points. These systems are designed to provide natural gas to the population, including domestic, industrial, and agricultural consumers. This study is aimed at developing methods for improving the calculation of gas distribution networks. The gas supply system should ensure an uninterrupted and safe gas supply to consumers that is easy to operate and provides the possibility of shutting down its individual elements for preventive, repair, and emergency recovery work. Therefore, this study presents a mathematical calculation method to find the optimal operating conditions for any gas network during the period of seasonal changes in thermal loads. This method demonstrates how the reliability of gas distribution systems and resistance to non-standard critical loads are affected by consumers based on the time of year, month, and day, and external factors such as outdoor temperature. The results in this study show that this method will enable the implementation of tools for testing various management strategies for the gas distribution network. [ABSTRACT FROM AUTHOR]

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm. [ABSTRACT FROM AUTHOR]

Because of mutations and selection, pathogens can manage to resist to drugs. However, the evolution of an asexual population (e.g., viruses, bacteria and cancer cells) depends on some external factors (e.g., antibiotic concentrations), and so understanding the impact of the environmental changes is an important issue. This paper is devoted to model this problem with a nonlocal diffusion PDE, describing the dynamics of such a phenotypically structured population, in a changing environment. The large-time behaviour of this model, with particular forms of environmental changes (linear or periodically fluctuations), has been previously developed. A new mathematical approach (limited to isotropic mutations) has been developed recently for this problem, considering a very general form of environmental variations, and giving an analytic description of the full trajectories of adaptation. However, recent studies have shown that an anisotropic mutation kernel can change the evolutionary dynamics of the population: some evolutive plateaus can appear. Thus the aim of this paper is to mix the two previous studies, with an anisotropic mutation kernel, and a changing environment. The main idea is to study a multivariate distribution of (2n) "fitness components". Its generating function solves a transport equation, and describes the distribution of fitness at any time. [ABSTRACT FROM AUTHOR]

An overview of the edge detection techniques based on partial differential equations (PDE) is presented in this work. Nonlinear anisotropic diffusion-based boundary extraction approaches, like the influential Perona-Malik model and some improved variants of it, are described first. Anisotropic diffusion-based detection schemes using the mean curvature motion and nonlinear PDE-based approaches combining anisotropic diffusion to the bilateral filter, are then disscused here. Some nonlinear reaction-diffusion based edge detection methods are described next. Variational edge detection solutions using the total variation (TV) regularization or combining the anisotropic diffusion to the TV-based models are then presented. Directional diffusion-based image edge extraction algorithms, are also disscused. Our own contributions in this computer vision domain are finally described. [ABSTRACT FROM AUTHOR]

Commemorative article for the 50th anniversary of the Black-Scholes mode presenting the derivation of the partial differential equation for pricing in the context of a continuous-time market model. The use of a physically-informed neural network (PINN) is proposed as a resolution method, as a novel technique in the field of scientific machine learning, which allows solving these types of equations without the need for a large amount of training data. The article includes the implementation of the method and the valuation results for the case of European call options. [ABSTRACT FROM AUTHOR]

The aim of the manuscript is to study new optical soliton solutions of the Kundu–Mukherjee–Naskar (KMN) equation via the ( G ′ / G , 1 / G )-expansion technique and the exponential rational function (ERF) procedure. The results are produced under the constraint conditions, and their graphical representation highlights them. These discoveries might aid in the comprehension of intricate nonlinear phenomena and oceanography. Studying the investigated equation in this paper is very important in explaining oceanographic phenomena such as rogue waves' ocean currents. [ABSTRACT FROM AUTHOR]

A numerical method for solving one-dimensional (1D) parabolic convection–diffusion equation is provided. We consider the finite difference formulas with five points to obtain a numerical method. The proposed method converts the given equation, domain, and time interval into a discrete form. The numerical values of the solution are approximated by solving algebraic equations containing finite differences and values at these discrete points. The consistency, stability and convergence are investigated. On the other hand, some numerical examples illustrate the validity and applicability of the method. Finally, the numerical results are compared with the finite difference scheme's three points. [ABSTRACT FROM AUTHOR]

In this paper, we present a new complex double integral transform namely "Complex Double Sadik Transform", to solve general linear partial differential equations. Several functions are used (applied) to show the usefulness of this new double transformation. [ABSTRACT FROM AUTHOR]

Surface reconstruction is an important part in automotive styling design. Existing reconstruction methods mainly rely on the proficiency of digital modelers who manually modify the surface shape to approximate the scanned data. Apart from manual modifications, the reconstructed surfaces cannot well reflect the design intent of designers since the feature curves of clay models have not been preserved accurately. In this paper, we propose a partial differential equation (PDE) based surface reconstruction method to analytically generate optimal surfaces with Cn continuity under the constraint of the feature curves. The proposed method accurately preserves automotive feature curves and achieves automatic reconstruction of Class-A surfaces without time-consuming manual work. The effectiveness of the proposed method is demonstrated by a number of experiments that reconstruct main parts of automotive exteriors. [ABSTRACT FROM AUTHOR]

In this study, we expanded the partial differential equation framework to which fractalfractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed. [ABSTRACT FROM AUTHOR]

In this paper, the differential quadrature and the finite difference combined method (DQ-FDM) was applied to solve the three-dimensional Burger's equation in the determination of the 3D velocity of the flow; so that spatial terms were discretized by the differential quadrature method, and the temporal term was discretized by the finite difference method, and the resulting nonlinear equations were solved using the Newton–Raphson method. All variables were considered as dimensionless in this equation. The solution results were compared with solution results of the two-dimensional equation in the two other numerical methods available in the literature which provided an acceptable accuracy. Also, the results of the mentioned numerical method were compared with those of the fully implicit finite difference method that was solved for larger than or equal viscosities of 0.1. The results showed that by increasing time and viscosity, the longitudinal, depth and transverse velocities were decreased. The occurrence of the upward flow was observed especially in the υ = 0.05 in the close of the bed, end of the length and width that in the presence of very fine particles of the clay and silt shows suspension of these particles in some spaces. The position of the longitudinal, depth and transverse velocities in the plan for the passing plates through the section depth for different viscosities and times showed that by increasing viscosity and time, the position of the maximum velocities became closer to the middle of the section width. Also, stream lines were plotted in all of sections and then analyzed. [ABSTRACT FROM AUTHOR]

Single-cell sequencing technologies have revolutionized molecular and cellular biology and stimulated the development of computational tools to analyze the data generated from these technology platforms. However, despite the recent explosion of computational analysis tools, relatively few mathematical models have been developed to utilize these data. Here we compare and contrast two cell state geometries for building mathematical models of cell state-transitions with single-cell RNA-sequencing data with hematopoeisis as a model system; (i) by using partial differential equations on a graph representing intermediate cell states between known cell types, and (ii) by using the equations on a multi-dimensional continuous cell state-space. As an application of our approach, we demonstrate how the calibrated models may be used to mathematically perturb normal hematopoeisis to simulate, predict, and study the emergence of novel cell states during the pathogenesis of acute myeloid leukemia. We particularly focus on comparing the strength and weakness of the graph model and multi-dimensional model.

In a recent study, an evolution equation is found for waves' behavior at far-field with relaxation mode of molecules. An analytical technique was used to solve this evolution problem, which is a generalized Burger equation. The analytical approach has limitations and requires a very accurate initial guess by a trial method. In this paper, the evolution equation for one-dimensional planar, cylindrical, and spherical flow in the presence of relaxation mode is solved using a collocation approach with a cubic B-spline function. The numerical results are graphed and compared with the exact solution for planar flow. The obtained numerical results match the exact solution quite well and show that the technique is quite reliable and can deal with the nonlinearity involved in the present problem. Results have also been obtained for cylindrical and spherical flow at the far-field. The obtained numerical results show that the present approach with the cubic B-spline function works well and accurately. Fourier stability analysis is used to investigate the stability of the cubic B-spline collocation method.

In this work, we propose an end-to-end adaptive sampling framework based on deep neural networks and the moving mesh method (MMPDE-Net), which can adaptively generate new sampling points by solving the moving mesh PDE. This model focuses on improving the quality of sampling points generation. Moreover, we develop an iterative algorithm based on MMPDE-Net, which makes sampling points distribute more precisely and controllably. Since MMPDE-Net is independent of the deep learning solver, we combine it with physics-informed neural networks (PINN) to propose moving sampling PINN (MS-PINN) and show the error estimate of our method under some assumptions. Finally, we demonstrate the performance improvement of MS-PINN compared to PINN through numerical experiments of four typical examples, which numerically verify the effectiveness of our method. [ABSTRACT FROM AUTHOR]

The intrinsic flexibility of integrated electricity and gas systems (IEGSs) hinges on the gas flow dynamics dictated by partial differential equations (PDEs). However, conventional numerical approaches for PDEs grapple with inefficiencies and inaccuracies for operational analysis, chiefly due to the discretization of PDE invoked. For the first time, this study pioneers an analytical methodology for IEGS's analysis, effectively eliminating the pitfalls of discretization in solving gas flow dynamics. Based on this, the dynamic optimal energy flow problem in IEGS is reformulated in continuous space, significantly simplifying complex models while bolstering analytical precision. Case studies affirm the remarkable advantages of the proposed analytical approach. Under comparable scenarios, the proposed method improves computational efficiency dozens of times and exhibits concurrently heightened levels of accuracy. • The analytical expression of gas flow dynamics without approximation or discretization is developed. • The dynamic optimal energy flow of IEGS is reformulated in continuous space for feasibility improvement. • The reformulated model improves the computational efficiency dozens of times and exhibits significant accuracy. [ABSTRACT FROM AUTHOR]

This paper explores the application of Physical-informed Neural Networks (PINNs) for reconstructing ship propeller wake fields, introducing a novel approach known as self-adaptive loss balanced physics-informed neural networks (LB-PINNs). It can enhance the accuracy of the reconstruction process by incorporating an adaptive weight to balance the loss term within the network. The initial sections of the paper present the foundational principles and framework of PINNs. To validate the efficacy of PINNs in solving partial differential equations, the well-known Burgers equation is applied and then these results with those obtained through LB-PINNs are compared. This comparative analysis highlights the superior performance of LB-PINNs in achieving accurate predictions. Moving forward, the open water characteristics of the KVLCC2 propeller are simulated by using computational fluid dynamics (CFD) software STAR CCM+, and the flow field information of the KVLCC2 propeller in open water is obtained. This simulation provides crucial flow field information, forming the basis for constructing a training sample set to train the neural network. The trained PINN and LB-PINN are used to infer approximate solutions of the governing equations at arbitrary time and space coordinates. The velocity and pressure distributions obtained by PINN and LB-PINN were compared with those simulated by STAR CCM+. The results confirm the applicability of both PINN and LB-PINN in the reconstruction of propeller wake fields, with LB-PINN demonstrating superior performance. • A novel method for solving governing differential equations of fluid mechanics based on LB-PINN is introduced. • LB-PINN's adaptive weight balances the loss term, reducing mean square error by an order of magnitude compared to PINN. • Sparse numerical simulation data suffices for LB-PINN training, enabling ship propeller wake reconstruction. • Comparing flow field reconstructions using PINN, LB-PINN, and CFD shows LB-PINN's validity and accuracy. [ABSTRACT FROM AUTHOR]

In this study, a boundary controller based on a peak detector system has been designed to reduce vibrations in the cable–tip–mass system. The control procedure is built upon a recent modification of the controller, incorporating a non-symmetric peak detector mechanism to enhance the robustness of the control design. The crucial element lies in the identification of peaks within the boundary input signal, which are then utilized to formulate the control law. Its mathematical representation relies on just two tunable parameters. Numerical experiments have been conducted to assess the performance of this novel approach, demonstrating superior efficacy compared to the boundary damper control, which has been included for comparative purposes.

We study a system of M particles with jump dynamics on a network of N sites. The particles can exist in two states, active or inactive. Only the former can jump. The state of each particle depends on its position. A given particle is inactive when it is at a given site, and active when it moves to a change site. Indeed, each sleeping particle activates at a rate λ>0, leaves its initial site, and moves for an exponential random time of parameter μ>0 before uniformly landing at a site and immediately returning to sleep. The behavior of each particle is independent of that of the others. These dynamics conserve the total number of particles; there is no limit on the number of particles at a given site. This system can be represented by a continuous-time Pólya urn with M balls where the colors are the sites, with an additional color to account for particles on the move at a given time t. First, using this Pólya interpretation for fixed M and N, we obtain the average number of particles at each site over time and, therefore, those on the move due to mass conservation. Secondly, we consider a large system in which the number of particles M and the number of sites N grow at the same rate, so that the M/N ratio tends to a scaling constant α>0. Using the moment-generating function technique added to some probabilistic arguments, we obtain the long-term distribution of the number of particles at each site.

The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the partial differential equation, expressed as an expectation value for Brownian motion. This paper demonstrates that the Feynman–Kac formula does not produce a unique solution but instead carries infinitely many solutions to the corresponding partial differential equation.

The main goal of this article is to obtain the existence of solutions for a nonlinear system of a coupled Petrovsky–Petrovsky system in the presence of infinite memories under minimal assumptions on the functions g1,g2 and φ1,φ2. Here, g1,g2 are relaxation functions and φ1,φ2 represent the sources. Also, a general decay rate for the associated energy is established. Our work is partly motivated by recent results, with a necessary modification imposed by the nature of our problem. In this work, we limit our results to studying the system in a bounded domain. The case of the entire domain Rn requires separate consideration. Of course, obtaining such a result will require not only serious technical work but also the use of new techniques and methods. In particular, one of the most significant points in achieving this goal is the use of the perturbed Lyapunov functionals combined with the multiplier method. To the best of our knowledge, there is no result addressing the linked Petrovsky–Petrovsky system in the presence of infinite memory, and we have overcome this lacune.

In this research paper, the authors aim to establish a novel algorithm in the finite difference method (FDM). The novel idea is proposed in the mesh generation process, the process to generate random grids. The FDM over a randomly generated grid enables fast convergence and improves the accuracy of the solution for a given problem; it also enhances the quality of precision by minimizing the error. The FDM involves uniform grids, which are commonly used in solving the partial differential equation (PDE) and the fractional partial differential equation. However, it requires a higher number of iterations to reach convergence. In addition, there is still no definite principle for the discretization of the model to generate the mesh. The newly proposed method, which is the SM method, employed randomly generated grids for mesh generation. This method is compared with the uniform grid method to check the validity and potential in minimizing the computational time and error. The comparative study is conducted for the first time by generating meshes of different cell sizes, i.e., 10×10,20×20,30×30,40×4010\times 10,\hspace{.25em}20\times 20,\hspace{.25em}30\times 30,\hspace{.25em}40\times 40 using MATLAB and ANSYS programs. The two-dimensional PDEs are solved over uniform and random grids. A significant reduction in the computational time is also noticed. Thus, this method is recommended to be used in solving the PDEs.

The basic objective of this paper is to utilize the factorization technique method to derive several properties such as, shift operators, recurrence relation, differential, integro-differential, partial differential expressions for Gould-Hopper-Frobenius-Genocchi polynomials, which can be utilized to tackle some new issues in different areas of science and innovation.

In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's conjecture).

Despite advancements in high-precision segmentation technology for computed tomographic angiography (CTA)-based cardiac wall segmentation, the accurate detection of the endocardial (Endo) and epicardial (Epi) boundaries remains a prerequisite for automated measurements of the cardiac wall thickness (WT). We proposed a novel algorithm for automated three-dimensional (3D) atrial WT (AWT) measurements, including an automatic Endo-Epi boundary detection. We detected the boundaries that were topologically indistinguishable due to an open geometry at the anatomical boundaries using the combined Convex hull and Poisson solver methods. The Laplace equation for the WT measurement was solved by a partial differential equation combining the two detected boundaries of the myocardial wall. We verified the robustness of our algorithm in mask images of the atrial wall that were separated from the CTA images of 20 patients and a phantom model. The accuracy of the automatically detected Endo-Epi boundaries was acceptable as compared to that manually extracted from the phantom model (Dice coefficient = 0.979). The 3D AWTs calculated by the novel automated method from the CTA images obtained from 20 patients with atrial fibrillation had

We present a framework for modeling asset and portfolio dynamics, incorporating this information into portfolio optimization. We define drivers for asset and portfolio dynamics and their optimal selection. For this framework, we introduce the Commonality Principle, providing a solution for the optimal selection of portfolio drivers as the common drivers. Portfolio constituent dynamics are modeled by Partial Differential Equations, and solutions approximated with neural networks. Sensitivities with respect to the common drivers are obtained via Automatic Adjoint Differentiation. Information on asset dynamics is incorporated via sensitivities into portfolio optimization. Portfolio constituents are embedded into the space of sensitivities with respect to their common drivers, and a distance matrix in this space called the Sensitivity matrix is used to solve the convex optimization for diversification. The sensitivity matrix measures the similarity of the projections of portfolio constituents on a vector space formed by common drivers’ returns and is used to optimize for diversification on both idiosyncratic and systematic risks while adding directionality and future behavior information via returns dynamics. For portfolio optimization, we perform hierarchical clustering on the sensitivity matrix. The clustering tree is used for recursive bisection to obtain the weights. To the best of the author’s knowledge, this is the first time that sensitivities’ dynamics approximated with neural networks have been used for portfolio optimization. Secondly, that hierarchical clustering on a matrix of sensitivities is used to solve the convex optimization problem and incorporate the hierarchical information of these sensitivities. Thirdly, public and listed variables can be used to obtain maximum idiosyncratic and systematic diversification by means of the sensitivity space with respect to optimal portfolio drivers. We reach over-performance in many experiments with respect to all other out-of-sample methods for different markets and real datasets. We also include a recipe for the methodology to increase performance even further, and tackle the main issues in portfolio management such as regimes, non-stationarity, overfitting, and selection bias.

In engineering applications, flexible beam vibration control is an important issue. Although several researchers have discussed controlling beam vibration, there are few strategies for implementing it in actual applications. The passivity-based boundary control for suppressing flexible beam vibration was investigated in this paper. The controller was implemented using a moving base, and the beam model was an undamped shear beam. The control law was established using the storage function in the design technique. The finite-gain L2 - stability of the feedback control system was then proven. This method dealt directly with the PDE of the beam model with no model reduction. Because of the non-collocated measurement and actuation in many applications, the backstepping observer was required for state estimation. Since the controller was implemented at the end of the beam via a moving base, the beam domain remained intact. Therefore, the method is simple to apply in applications. With the use of the finite-difference approach, the PDEs were numerically solved. The controller's performance of the proposed control scheme was demonstrated using computer simulation.

Simple Summary: Osteosarcoma is the most common type of bone cancer seen in children to young adults with poor prognosis. To find effective treatments, it is crucial to understand the mechanism of the initiation and progression of the osteosarcoma tumors. In this paper, we introduce a PDE model for the progression of osteosarcoma tumors by considering the location of different cell types, including immune and cancer cells, in the tumors. We perform several simulations using the developed model to investigate the importance and role of the different immune cells' location in the growth of the tumors. The results show that the co-localization of macrophages and cancer cells promotes tumors' growth. Osteosarcoma is the most common malignant bone tumor in children and adolescents with a poor prognosis. To describe the progression of osteosarcoma, we expanded a system of data-driven ODE from a previous study into a system of Reaction-Diffusion-Advection (RDA) equations and coupled it with Biot equations of poroelasticity to form a bio-mechanical model. The RDA system includes the spatio-temporal information of the key components of the tumor microenvironment. The Biot equations are comprised of an equation for the solid phase, which governs the movement of the solid tumor, and an equation for the fluid phase, which relates to the motion of cells. The model predicts the total number of cells and cytokines of the tumor microenvironment and simulates the tumor's size growth. We simulated different scenarios using this model to investigate the impact of several biomedical settings on tumors' growth. The results indicate the importance of macrophages in tumors' growth. Particularly, we have observed a high co-localization of macrophages and cancer cells, and the concentration of tumor cells increases as the number of macrophages increases. [ABSTRACT FROM AUTHOR]

Piezoelectric devices belong to the most prominent examples of smart materials. They find their applications in sensors and actuators, e.g. in the context of ultrasonic applications, tomography, cavitation-based cleaning. Nowadays, the design of new piezoelectric devices is generally accompanied by a computer-aided design, i.e. some models are used to predict the mechanical–electrical coupling of the new products. The coupling is described by a set of second-order coupled partial differential equations. For the mechanical part, this system comprises the equation of motion for the mechanical displacement in three dimensions and for the electric part, an electrostatic potential equation is employed. Coupling terms and an additional Rayleigh damping approach ensure the validness of the model. In this work, we analyze the existence, uniqueness and regularity of the solutions to these equations and give a result concerning the long-term behavior. The assumptions mainly on the material parameters involved are quite natural and allow meaningful physical interpretation. [ABSTRACT FROM AUTHOR]

In this paper, the solution methodology of higher-order linear fractional partial deferential equations (FPDEs) as mentioned in eqs (1) and (2) below in Caputo definition relies on a new analytical method which is called the Laplace-residual power series method (L-RPSM). The main idea of our proposed technique is to convert the original FPDE in Laplace space, and then apply the residual power series method (RPSM) by using the concept of limit to obtain the solution. Some interesting and important numerical test applications are given and discussed to illustrate the procedure of our method, and also to confirm that this method is simple, understandable and very fast for obtaining the exact and approximate solutions (ASs) of FPDEs compared with other methods such as RPSM, variational iteration method (VIM), homotopy perturbation method (HPM) and Adomian decomposition method (ADM). The main advantage of the proposed method is its simplicity in computing the coefficients of terms of series solution by using only the concept of limit at infinity and not as the other well-known analytical method such as, RPSM that need to obtain the fractional derivative (FD) each time to determine the unknown coefficients in series solutions, and VIM, ADM, or HPM that need the integration operators which is difficult in fractional case. [ABSTRACT FROM AUTHOR]

This study investigates the vertical profile of streamwise (longitudinal) velocity in an unsteady and nonuniform open-channel turbulent flow. In contrast to the previous works, a direct derivation for velocity distribution starting from Reynolds-averaged Navier-Stokes (RANS) equation has been shown. Due to unavailability of several expressions for unsteady flow to solve the governing equation, a few assumptions have been made. The proposed model contains the effect of secondary current, which is generally present in all types of open-channel flow. The resulting partial differential equation has been solved numerically. Validation has been done by comparing the model with available experimental data and an existing analytical model. [ABSTRACT FROM AUTHOR]

Accurate knowledge of the natural frequencies and shapes of corresponding standing waves are essential for gaining deeper insight into the nature of response of coils to impulse excitations. Most of the previous analytical studies on coils assumed shape of standing waves as sinusoidal but numerical circuit analysis and measurements suggest otherwise. Hence, this paper revisits the classical standing wave phenomenon in coils to ascertain reasons for this discrepancy and thereafter extends it by analytically deriving the exact mode shape of standing waves for both neutral open/short conditions. For this, the coil is modeled as a distributed network of elemental inductances and capacitances while spatial variation of mutual inductance between turns is described by an exponential function. Initially, an elegant derivation of the governing partial differential equation for surge distribution is presented which is then analytically solved, perhaps for the first time, by the variable-separable method to find the complete solution (sum of time and spatial terms). Hyperbolic terms in spatial part of solution have always been neglected but are included here, thus, yielding the exact mode shapes. Voltage standing waves gotten from analytical solution are plotted and compared with simulation results on a 100-section ladder network. The same is measured on a large-sized single layer coil. So, it emerges that, even in single layer coils, shape of standing waves deviates considerably from being sinusoidal and this deviation depends on spatial variation of mutual inductance, capacitive coupling, and order of standing waves. [ABSTRACT FROM AUTHOR]

The stability theory is an important research area in the qualitative analysis of partial differential equations. The Hyers-Ulam stability for a partial differential equation has a very close exact solution to the approximate solution of the differential equation and the error is very small which can be estimated. This study examines Hyers-Ulam and Hyers-Ulam Rassias stability of second order partial differential equations. We present a new method for research of the Hyers-Ulam stability of partial differential equations with the help of the Mahgoub transform. The Mahgoub transform method is practical as a fundamental tool to demonstrate the original result on this study. Finally, we give an example to illustrate main results. Our findings make a contribution to the topic and complete those in the relevant literature. [ABSTRACT FROM AUTHOR]

This paper proposes a methodological framework that incorporates probabilistic facility disruption risks, last-mile customers travel path choices, and the induced traffic congestion near the facilities into the consideration of service facility location planning. The customers can be pedestrians, drones, or any autonomous vehicles that do not have to travel via fixed channels to access a service facility. Analytical models are developed to evaluate the expected performance of a facility location design across an exponential number of facility disruption scenarios. In each of these scenarios, customers travel to a functioning facility through a continuous space, and their destination and path choices under traffic equilibrium are described by a class of partial differential equation (PDE). A closed-form solution to the PDE is derived in an explicit matrix form, and this paper shows how the traffic equilibrium patterns across all facility disruption scenarios can be evaluated in a polynomial time. These new analytical results are then incorporated into continuous and discrete optimization frameworks for facility location design. Numerical experiments are conducted to test the computational performance of the proposed modeling framework. • Reliable facility location design under probabilistic facility disruption risks. • Congestion and continuous traffic equilibrium in a continuous space without guideways. • Analytical models to evaluate the expected performance of a facility location design. • System cost across an exponential number of disruption scenarios evaluated in a polynomial time. • Analytical results incorporated into continuous and discrete optimization frameworks. [ABSTRACT FROM AUTHOR]

In the field of state estimation for the lithium-ion battery (LIB), model-based methods (white box) have been developed to explain battery mechanism and data-driven methods (black box) have been designed to learn battery statistics. Both white box methods and black box methods have drawn much attention recently. As the combination of white box and black box, physics-informed machine learning has been investigated by embedding physic laws. For LIB state estimation, this work proposes a fractional-order recurrent neural network (FORNN) encoded with physics-informed battery knowledge. Three aspects of FORNN can be improved by learning certain physics-informed knowledge. Firstly, the fractional-order state feedback is achieved by introducing a fractional-order derivative in a forward propagation process. Secondly, the fractional-order constraint is constructed by a voltage partial derivative equation (PDE) deduced from the battery fractional-order model (FOM). Thirdly, both the fractional-order gradient descent (FOGD) and fractional-order gradient descent with momentum (FOGDm) methods are proposed by introducing a fractional-order gradient in the backpropagation process. For the proposed FORNN, the sensitivity of the added fractional-order parameters are analyzed by experiments under the federal urban driving schedule (FUDS) operation conditions. The experiment results demonstrate that a certain range of every fractional-order parameter can achieve better convergence speed and higher estimation accuracy. On the basis of the sensitivity analysis, the fractional-order parameter tuning rules have been concluded and listed in the discussion part to provide useful references to the parameter tuning of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Solving partial differential equations plays a vital role of numerical analysis in scientific and engineering fields such as computational fluid dynamics. Due to the multi-scale nature of physics and sensitivity to the quality of the discrete mesh, traditional numerical methods often require complex human-computer interaction and expensive meshing overhead, which limit their application to many realtime simulation and optimal design problems. This paper proposes an improved neural network-based methodforsolvingpartialdiferentialequations, named TaylorPINN. It utilizes the universal approximation theorem of neural networks and the function-fitting capability of Taylor formula, and provides a mesh-freenumericalsolvingprocess. Numerical experimental resultson Helmholtz, Klein-Gordon, and Navier-Stokesequations demonstrate that Taylor PINN is able to approximate the underlying mapping relationsbetweenthecoordinateinputsandquantitiesofinterest, yielding an a ccurateprediction result. Comparedwiththe widely used physics-informed neural network method, TaylorPINN improves the prediction accuracy by a factor of 3~20x across different numerical problems. [ABSTRACT FROM AUTHOR]

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species. [ABSTRACT FROM AUTHOR]

A system of ordinary differential equations of large dimension that models a multistage synthesis is considered. A new method for constructing an approximate solution of the Cauchy problem is proposed. The method is based on the established connections between the solutions of the system of differential equations, a delay equation, and a partial differential equation of parabolic type. [ABSTRACT FROM AUTHOR]

We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data u 01 ∈ L 2 and u 02 ∈ H − 1 + η for η > 0. [ABSTRACT FROM AUTHOR]

Reaction–diffusion problems are found in several fields, such as fluid dynamics, heat transfer and electromagnetism. An accurate and efficient numerical method to obtain a solution to this class of problems is still a thriving topic of research. In this article, a new promising numerical strategy to solve this type of problems is proposed. The main ideas are the use of the Crank–Nicolson method to discretize the time domain of the problem and the modification of the standard approach of the finite element method. In this modification, construction and evaluation of the linear system, which is obtained from the spatial discretization, are replaced by a variational inference problem on a Gaussian multivariate probability distribution, originated from the energy functional of the formulation. Then, the Gaussian belief propagation procedure is used to calculate the marginal mean of the distribution, which represents the approximated solution. GaBP is an iterative message-passing algorithm usually employed to solve distributed variational inference problems on graphical models. The present method is of great value, mostly in large scale systems, since it suppresses the linear system resolution, which is a high time-consuming process in standard methods. [ABSTRACT FROM AUTHOR]