Solution Of Partial Differential Equations Research Articles

Examples of dHYM connections in a variable background

Abstract We study deformed Hermitian Yang–Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application, we provide many new examples of dHYM connections coupled to a variable background Kähler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the Kähler–Yang–Mills system of Álvarez-Cónsul, Garcia-Fernandez, and García-Prada, and in this limit, our solutions converge smoothly to those constructed by Keller and Tønnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realization as B-branes, the small radius limit, and canonical representatives of complexified Kähler classes.

Exact solutions of classes of second order nonlinear partial differential equations reducible to first order

This paper illustrates the successful implementation of the method of variation of parameters in combination with the method of characteristics and other techniques to obtain exact solutions for a wide range of partial differential equations. The proposed approach reduces partial differential equations (PDEs) to first-order differential equations, referred to as classical equations, including Bernoulli, Ricatti, and Abel equations. In addition, the techniques proposed have the ability to produce precise solutions for nonlinear second order PDEs. For each PDE class, the method's effectiveness is demonstrated through illustrative examples.

A new analytical approach for the local radial point interpolation discretisation in space and applications to high-order in time schemes for two-dimensional fractional PDEs

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity.

Error homogenization in physics-informed neural networks for modeling in manufacturing

Physics-informed neural networks (PINNs) have demonstrated effectiveness in solving partial differential equations (PDEs) associated with manufacturing scenarios, due to their physically interpretable training logic. One limitation has been that PINNs often exhibit heterogenous error maps and high-error “hot spots” throughout the solution domain, which reduce not only the solutions’ accuracy but also their overall consistency with the physical laws. This study addresses this limitation by presenting an efficient and error-aware PINN ensembling technique for error homogenization in solving manufacturing problems. Specifically, a PINN is first established by constraining its training process through a manufacturing specific PDE and corresponding boundary conditions to ensure physical consistency. Next, the loss landscape in the neighborhood of three PINNs trained with varying network parameter initialization is sampled to generate a PINN ensemble. Finally, the outputs of the ensemble members are combined through an inverse error-weighted average to yield the prediction of the PDE solution. Evaluation using the Allen-Cahn PDE, which describes phase separation in the solidification of metallic alloys, shows that the developed method reduces the average prediction error by 63% and error standard deviation by 30% across the solution space, demonstrating its effectiveness for PINN error reduction and homogenization. Additionally, the method has also demonstrated 96% reduction in the computational time as compared to conventional ensembling methods.

Enhancing accuracy of physically informed neural networks for nonlinear Schrödinger equations through multi-view transfer learning

In recent years, significant research efforts have been dedicated to developing solutions for nonlinear partial differential equations (PDEs) with applications in physics. Among these equations, the NonLinear Schrödinger (NLS) equation plays a pivotal role in modern nonlinear science, governing the behavior of optical solitons and serving as a crucial model. However, the practical application of these elegant PDE models is hindered by the challenge of high dimensionality and computational cost. To address these challenges, we introduce a novel approach to enhance the training accuracy of physically informed neural networks (PINNs) for the NLS equation. Leveraging the concept of multi-view transfer learning, we utilize a pre-trained NLS model that considers different initial–boundary conditions as multiple views, enabling exploration of their contributions to the overall understanding and solution of the NLS equation. Additionally, we investigate the influence of different loss functions on PINN training results and find that employing the Huber Loss function yields superior fitting outcomes compared to traditional PINNs that use mean squared error (MSE) loss. We demonstrate significant improvements in precision and accuracy over traditional PINNs. Extensive experiments validate the effectiveness of our proposed optimization techniques. This work paves a novel way for solving complex physical systems using data-driven approaches.

Методика определения минимально допустимой толщины обода цельнокатаных колес грузовых вагонов в эксплуатации

Purpose: Determination of the minimum allowable thickness of the rim of solid-rolled wheels of freight cars depending on the design and axial load. Methods: Methods of elasticity theory, methods of numerical solution of partial differential equations (finite element method) are used. Results: The minimum allowable thicknesses of the rims of solid-rolled wheels of freight cars have been determined, as well as the values of fatigue resistance factors for wheels manufactured according to drawing 13.45.1043-01A, 13.45.1011-143, 13.45.1399-207 with a rim thickness of 30 to 16 mm and various axial load. The calculations showed that for the wheels manufactured according to drawing 13.45.1043-01A, 13.45.1399-207, the minimum permissible rim thickness can be reduced to 16 mm. Practical significance: The dependence of the fatigue resistance coefficient on the rim thickness and axial load with an accuracy of up to 15% is obtained, as well as certain minimum permissible thicknesses of the rims of solid-rolled wheels will increase their service life.

Method of separation of variables and exact solution of time fractional nonlinear partial differential and differential-difference equations

Method of separation of variables and exact solution of time fractional nonlinear partial differential and differential-difference equations

A hybrid technique based on Lucas polynomials for solving fractional diffusion partial differential equation

This paper presents a new numerical technique to approximate solutions of diffusion partial differential equations with Caputo fractional derivatives. We use a spectral collocation method based on Lucas polynomials for time fractional derivatives and a finite difference scheme in space. Stability and error analyses of the proposed technique are established. To demonstrate the reliability and efficiency of our new technique, we applied the method to a number of examples. The new technique is simply applicable, and the results show high efficiency in calculation and approximation precision.

Besov regularity of inhomogeneous parabolic PDEs

We study the regularity of solutions of parabolic partial differential equations with inhomogeneous boundary conditions on polyhedral domains Dsubset mathbb {R}^3 in the specific scale B^{alpha }_{tau ,tau }, frac{1}{tau }=frac{alpha }{3}+frac{1}{p} of Besov spaces. The regularity of the solution in this scale determines the order of approximation that can be achieved by adaptive numerical schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our results are in good agreement with the forerunner (Dahlke and Schneider in Anal Appl 17:235–291, 2019), where parabolic equations with homogeneous boundary conditions were investigated.

THE USE OF MECHANICAL FILTER MODELS IN THE ANALYSIS OF FORMING AND COMPACTION PROCESSES OF FORMATION AND COMPACTION OF BUILDING/CONCRETE MIXTURES BY VIBRATING FIELD

The paper describes the use of various types of mechanical filter models, which are used for the analysis of the processes of formation and compaction of the construction/concrete mixtures of building/concrete mixtures by means of vibrating fields. The values of resonant frequencies and equivalent masses for different resonators modeling the propagation in the latter of vibrating-wave formations have been established. The analysis of the influence of a vibrating field on the processes of formation and compaction of concrete/concrete mixtures in this study is based on the methods of mathematical physics, classical variation calculus, physics of oscillations and waves and methodology of solution of ordinary differential equations and partial differential equations. The conditions and main integral characteristics of resonance phenomena, the possibility of occurrence of which is conditioned by: 1) the geometry of the initial boundary-edge problem (it is The so-called "geometric resonances" of the considered system with distributed parameters simulating the mixture to be processed); 2) the working rheological model of the mixture involved in the study (these are the so-called "rheological resonances"). The approach developed and scientifically substantiated in this work allows us to establish the main parameters and opportunities for the use of energy-saving modes of operation of vibration systems intended for the formation and vibration compaction of the above mixtures. The results obtained in the work The results obtained can be further used to clarify and Improvement of existing engineering methods of calculation of vibration systems for the formation and compaction of concrete/concrete mixtures in order to optimize the operating modes of their functioning both at the design stage and in the modes of real operation.

Stress state and waves in the lithospheric plate simulation: A 3rd generation AI architecture

Natural disasters present ongoing risks to human life and the global economy, with climate change and environmental factors exacerbating these threats. This article introduces an innovative approach to earthquake prediction and modeling, utilizing a combination of modern mathematical techniques, data-driven methodologies, and artificial intelligence (AI). By integrating the finite element method and wavelet collocation method, our approach enables the solution of complex partial differential equations involving fractional derivatives. This integration provides a more accurate prediction of earthquake behavior by leveraging AI analysis of the resulting numerical solutions.Our advanced modeling technique enables three-dimensional earthquake modeling, surpassing the limitations of traditional methods and offering unprecedented insights. The comprehensive AI approach shows great potential for improving our understanding of earthquake behavior, facilitating the design of earthquake-resistant structures, and effectively mitigating the devastating impacts of earthquakes.This research significantly contributes to the field of disaster risk reduction and offers practical applications for both researchers and practitioners. By combining mathematical rigor, data-driven insights, and the power of AI, our next-generation approach opens new avenues for enhancing our preparedness and response to seismic events.

Complex Analytic Solutions of Certain Partial Differential Equations

Complex Analytic Solutions of Certain Partial Differential Equations

A stable analytical expression for magnetotelluric/radiomagnetotelluric impedances in anisotropic layered Earth models

Previous closed-form solutions of impedances on layered Earth models were only available for the anisotropic conductivity case under the plane wave electromagnetic incidence. Here, we have developed a numerically stable impedance formula for both magnetotelluric and radiomagnetotelluric problems in layered media with arbitrarily anisotropic conductivity and dielect permittivity. Maxwell equations are first transformed into a set of coupled second-order partial differential equations with the horizontal components of electric field as the variables. Then, the recursive formula of surface impedance is obtained by using the general solutions of the second-order partial differential equations and the tangential continuity condition of electromagnetic fields at each layer interface. To avoid the numerical overflow in both the nominator and denominator of the recursive impedance formula due to large layer thickness or high working frequency, an artificial exponential function with positive exponential factor is used to regularize both the nominator and denominator. It leads numerically stable recursive formula for any layer thickness and working frequency. At the end, a four layered magnetotelluric model with anisotropic conductivities is adopted to validate the newly derived analytical formula and another two synthetic layered Earth models are used to study the effects of conductive anisotropy and dielectric anisotropy on radiomagnetotelluric responses.

The modified generalized Kudryashov method for nonlinear space–time fractional partial differential equations of Schrödinger type

This paper presents a modified version of the generalized Kudryashov method aimed at obtaining exact solutions for fractional partial differential equations of Schrödinger type. Firstly, the fractional partial differential equations are transformed into a set of ordinary differential equations using the fractional complex transform. Subsequently, a modified version of the generalized Kudryashov method is employed to solve these equations and the obtained results are utilized to derive the traveling wave solutions for fractional partial differential equations of the Schrödinger type. Finally, the graphs resulting from the traveling wave solution are examined, encompassing the analysis of kink waves and singular kink waves.

A New Hybrid Optimal Auxiliary Function Method for Approximate Solutions of Non-Linear Fractional Partial Differential Equations

This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville’s fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method’s effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques.

A REPRODUCING KERNEL METHOD FOR SOLVING SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.

Solving nonlinear soliton equations using improved physics-informed neural networks with adaptive mechanisms

Partial differential equations (PDEs) are important tools for scientific research and are widely used in various fields. However, it is usually very difficult to obtain accurate analytical solutions of PDEs, and numerical methods to solve PDEs are often computationally intensive and very time-consuming. In recent years, Physics Informed Neural Networks (PINNs) have been successfully applied to find numerical solutions of PDEs and have shown great potential. All the while, solitary waves have been of great interest to researchers in the field of nonlinear science. In this paper, we perform numerical simulations of solitary wave solutions of several PDEs using improved PINNs. The improved PINNs not only incorporate constraints on the control equations to ensure the interpretability of the prediction results, which is important for physical field simulations, in addition, an adaptive activation function is introduced. By introducing hyperparameters in the activation function to change the slope of the activation function to avoid the disappearance of the gradient, computing time is saved thereby speeding up training. In this paper, the mKdV equation, the improved Boussinesq equation, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the p-gBKP equation are selected for study, and the errors of the simulation results are analyzed to assess the accuracy of the predicted solitary wave solution. The experimental results show that the improved PINNs are significantly better than the traditional PINNs with shorter training time but more accurate prediction results. The improved PINNs improve the training speed by more than 1.5 times compared with the traditional PINNs, while maintaining the prediction error less than 10−2 in this order of magnitude.

An asymmetric probability density function

A probability density or distribution function of turbulence has been thought to be symmetric due to the symmetry of the partial differential equations from the first principles. However, the experimental data have been shown otherwise by a so-called Taylor correlation function, and this is an unresolved issue. A recent study shows that this probability density function can be asymmetric analytically by introducing negative eigenvalues. Here, in this work, we show the mathematical basis for this asymmetry, although the partial differential equations follow the symmetry of the Lie groups. We also demonstrate a complete solution of partial differential equations, including the exponential terms and negative eigenvalues, which plays a vital part in transient phenomena. Our analysis shows that the asymmetry is from the partition of velocities of the same or opposing direction, not from the negative eigenvalues. Fundamentally, the loss of rotational symmetry is caused by the exponential terms for the transient solution, which we demonstrate by the derivation of the complete solution. The new correlation produces excellent agreement with the experimental data. The universality and limitations of the correlation function are discussed, and through the parameter study, the variations and statistical nature of the probability function are clarified. The asymmetry probability function should have wider applicability than the symmetric Gaussian distribution, which is the special case of the asymmetry probability function.

Neural-Network-Assisted Finite Difference Discretization for Numerical Solution of Partial Differential Equations

A neural-network-assisted numerical method is proposed for the solution of Laplace and Poisson problems. Finite differences are applied to approximate the spatial Laplacian operator on nonuniform grids. For this, a neural network is trained to compute the corresponding coefficients for general quadrilateral meshes. Depending on the position of a given grid point x0 and its neighbors, we face with a nonlinear optimization problem to obtain the finite difference coefficients in x0. This computing step is executed with an artificial neural network. In this way, for any geometric setup of the neighboring grid points, we immediately obtain the corresponding coefficients. The construction of an appropriate training data set is also discussed, which is based on the solution of overdetermined linear systems. The method was experimentally validated on a number of numerical tests. As expected, it delivers a fast and reliable algorithm for solving Poisson problems.

How Can (or Why Should) Process Engineering Aid the Screening and Discovery of Solid Sorbents for CO2 Capture?

ConspectusAdsorption using solid sorbents is emerging as a serious contender to amine-based liquid absorption for postcombustion CO2 capture. In the last 20+ years, significant efforts have been invested in developing adsorption processes for CO2 capture. In particular, significant efforts have been invested in developing new adsorbents for this application. These efforts have led to the generation of hundreds of thousands of (hypothetical and real) adsorbents, e.g., zeolites and metal-organic frameworks (MOFs). Identifying the right adsorbent for CO2 capture remains a challenging task. Most studies are focused on identifying adsorbents based on certain adsorption metrics. Recent studies have demonstrated that the performance of an adsorbent is intimately linked to the process in which it is deployed. Any meaningful screening should thus consider the complexity of the process. However, simulation and optimization of adsorption processes are computationally intensive, as they constitute the simultaneous propagation of heat and mass transfer fronts; the process is cyclic, and there are no straightforward design tools, thereby making large-scale process-informed screening of sorbents prohibitive.This Account discusses four papers that develop computational methods to incorporate process-based evaluation for both bottom-up (chemistry to engineering) screening problems and top-down (engineering to chemistry) inverse problems. We discuss the development of the machine-assisted adsorption process learning and emulation (MAPLE) framework, a surrogate model based on deep artificial neural networks (ANNs) that can predict process-level performance by considering both process and material inputs. The framework, which has been experimentally validated, allows for reliable, process-informed screening of large adsorbent databases. We then discuss how process engineering tools can be used beyond adsorbent screening, i.e., to estimate the practically achievable performance and cost limits of pressure vacuum swing adsorption (PVSA) processes should the ideal bespoke adsorbent be made. These studies show what conditions stand-alone PVSA processes are attractive and when they should not be considered. Finally, recent developments in physics-informed neural networks (PINNS) enable the rapid solution of complex partial differential equations, providing tools to potentially identify optimal cycle configurations. Ultimately, we provide areas where further developments are required and emphasize the need for strong collaborations between chemists and chemical engineers to move rapidly from discovery to field trials, as we do not have much time to fulfill commitments to net-zero targets.