Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

Application of Image Denoising Method Based on Two-Way Coupling Diffusion Equation in Public Security Forensics

This paper uses the web live broadcast and on-demand platform based on the B/S architecture as the application side and designs a video image forensic system that can meet multiple police types and multiple application scenarios. The system uses mobile phones as the video image capture terminal to solve the problem of rapid response and concealment and uses 5G communication technology as the transmission medium to solve the problem of device mobility and link maintenance. The problem of diversification of the use and application modes of multiple police types is solved; the video image evidence is managed in a centralized storage, audit, and export method, and the security and authenticity of the evidence are solved. While the system realizes a series of functions such as the collection, transmission, storage, and application of video image evidence, it also realizes the application-side video image live broadcast function according to actual work needs and solves the large-scale case command and decision-making problem that has been plagued by public security organs. In order to remove the noise in the public security forensic images and to smooth the noise while retaining the details of the image, this paper proposes a denoising algorithm based on the two-way coupling diffusion equation. By improving the second-order partial differential equation, a new diffusion function with better diffusion effect than the original model is constructed. We combined the adaptive edge threshold and stop criterion to establish a new denoising algorithm model, which can get better denoising results. When the noise level is low, the PSNR value and SSIM value of several denoising methods are relatively ideal, and the result is at a higher level, the denoising picture effect is better, and there is no obvious incomplete noise removal or detail problems. As the noise level increases, the denoising results will gradually decrease, and the effects will also vary to different degrees. When the noise intensity increases, visually, it can be clearly seen that the two-way coupled diffusion equation and DnCNN have better denoising effects. When the noise level is high, the two-way coupled diffusion equation network is used to use the clear image and the denoised image for indistinguishable calculation. The method in this paper almost retains all the texture details in the clear image, and there are almost no artifacts and images. On the other hand, the color of the image after denoising by the method in this paper is more vivid, and it is closer to the target picture in terms of picture definition and tone, the denoising effect is ideal, and the generated image has a higher degree of restoration. Compared with the residual GAN, the two-way coupling diffusion equation network converges faster and the network performance is improved.

Factorized One-Way Wave Equations

The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- and backward-traveling wave results. Therefore, the Cauchy equation has to be regarded as a two-way wave equation, whose inherent directional ambiguity leads to irregular phantom effects in the numerical finite element (FE) and finite difference (FD) calculations. For seismic applications, a huge number of methods have been developed to reduce these disturbances, but none of these attempts have prevailed to date. However, a priori factorization of the longitudinal wave equation for inhomogeneous media eliminates the above-mentioned ambiguity, and the resulting one-way equations provide the definition of the wave propagation direction by the geometric position of the transmitter and receiver.

Multivector Fields of Noether Symmetries in the Lagrangian Formalism and Belinfante Tensor

Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar\'e-Cartan form.\\ The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.

Noise Data Removal and Image Restoration Based on Partial Differential Equation in Sports Image Recognition Technology

With the rapid development of image processing technology, the application range of image recognition technology is becoming more and more extensive. Processing, analyzing, and repairing graphics and images through computer and big data technology are the main methods to obtain image data and repair image data in complex environment. Facing the low quality of image information in the process of sports, this paper proposes to remove the noise data and repair the image based on the partial differential equation system in image recognition technology. Firstly, image recognition technology is used to track and obtain the image information in the process of sports, and the fourth-order partial differential equation is used to optimize and process the image. Finally, aiming at the problem of low image quality and blur in the transmission process, denoising is carried out, and image restoration is studied by using the adaptive diffusion function in partial differential equation. The results show that the research content of this paper greatly improves the problems of blurred image and poor quality in the process of sports and realizes the function of automatically tracking the target of sports image. In the image restoration link, it can achieve the standard repair effect and reduce the repair time. The research content of this paper is effective and applicable to image processing and restoration.

Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.

PDE Surface-Represented Facial Blendshapes

Partial differential equation (PDE)-based geometric modelling and computer animation has been extensively investigated in the last three decades. However, the PDE surface-represented facial blendshapes have not been investigated. In this paper, we propose a new method of facial blendshapes by using curve-defined and Fourier series-represented PDE surfaces. In order to develop this new method, first, we design a curve template and use it to extract curves from polygon facial models. Then, we propose a second-order partial differential equation and combine it with the constraints of the extracted curves as boundary curves to develop a mathematical model of curve-defined PDE surfaces. After that, we introduce a generalized Fourier series representation to solve the second-order partial differential equation subjected to the constraints of the extracted boundary curves and obtain an analytical mathematical expression of curve-defined and Fourier series-represented PDE surfaces. The mathematical expression is used to develop a new PDE surface-based interpolation method of creating new facial models from one source facial model and one target facial model and a new PDE surface-based blending method of creating more new facial models from one source facial model and many target facial models. Some examples are presented to demonstrate the effectiveness and applications of the proposed method in 3D facial blendshapes.

Gradient plasticity in isotropic solids

We discuss a framework for the description of gradient plasticity in isotropic solids based on the Riemannian curvature derived from a metric induced by plastic deformation. This culminates in a flow rule in the form of a fourth-order partial differential equation for the plastic strain rate, in contrast to the second-order flow rules that have been proposed in alternative treatments of gradient plasticity in isotropic solids.

Photovoltaic Energy All-Day and Intra-Day Forecasting Using Node by Node Developed Polynomial Networks Forming PDE Models Based on the L-Transformation

Forecasting Photovoltaic (PV) energy production, based on the last weather and power data only, can obtain acceptable prediction accuracy in short-time horizons. Numerical Weather Prediction (NWP) systems usually produce free forecasts of the local cloud amount each 6 h. These are considerably delayed by several hours and do not provide sufficient quality. A Differential Polynomial Neural Network (D-PNN) is a recent unconventional soft-computing technique that can model complex weather patterns. D-PNN expands the n-variable kth order Partial Differential Equation (PDE) into selected two-variable node PDEs of the first or second order. Their derivatives are easy to convert into the Laplace transforms and substitute using Operator Calculus (OC). D-PNN proves two-input nodes to insert their PDE components into its gradually expanded sum model. Its PDE representation allows for the variability and uncertainty of specific patterns in the surface layer. The proposed all-day single-model and intra-day several-step PV prediction schemes are compared and interpreted with differential and stochastic machine learning. The statistical models are evolved for the specific data time delay to predict the PV output in complete day sequences or specific hours. Spatial data from a larger territory and the initially recognized daily periods enable models to compute accurate predictions each day and compensate for unexpected pattern variations and different initial conditions. The optimal data samples, determined by the particular time shifts between the model inputs and output, are trained to predict the Clear Sky Index in the defined horizon.