Journal articles: 'Differential equations, Nonlinear Data processing'

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Enciso-Salas, Luis, Gustavo Pérez-Zuñiga, and Javier Sotomayor-Moriano. "Fault Diagnosis via Neural Ordinary Differential Equations." Applied Sciences 11, no. 9 (April 22, 2021): 3776. http://dx.doi.org/10.3390/app11093776.

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Implementation of model-based fault diagnosis systems can be a difficult task due to the complex dynamics of most systems, an appealing alternative to avoiding modeling is to use machine learning-based techniques for which the implementation is more affordable nowadays. However, the latter approach often requires extensive data processing. In this paper, a hybrid approach using recent developments in neural ordinary differential equations is proposed. This approach enables us to combine a natural deep learning technique with an estimated model of the system, making the training simpler and more efficient. For evaluation of this methodology, a nonlinear benchmark system is used by simulation of faults in actuators, sensors, and process. Simulation results show that the proposed methodology requires less processing for the training in comparison with conventional machine learning approaches since the data-set is directly taken from the measurements and inputs. Furthermore, since the model used in the essay is only a structural approximation of the plant; no advanced modeling is required. This approach can also alleviate some pitfalls of training data-series, such as complicated data augmentation methodologies and the necessity for big amounts of data.

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Lazarev, Alexander. "THE TECHNOLOGY OF WINTER CONCRETING OF MONOLITHIC FRAME STRUCTURES WITH SUBSTANTIATION OF HEAT TREATMENT MODES BY SOLUTIONS OF THE DIFFERENTIAL EQUATION OF THERMAL CONDUCTIVITY OBTAINED BY THE METHOD OF GROUP ANALYSIS." International Journal for Computational Civil and Structural Engineering 17, no. 4 (December 26, 2021): 115–22. http://dx.doi.org/10.22337/2587-9618-2021-17-4-115-122.

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An innovative method for calculating thermal fields inside monolithic structures has been developed, based on the use and analysis of nonlinear differential equations. The innovativeness of the method lies in the approach to the analysis of nonlinear physical processes using nonlinear differential equations. Thanks to the method of group analysis, 13 expressions are obtained from complex mathematical equations, which are easy to use and depend on several empirical coefficients. It is assumed that this calculation method is a priori more accurate than the existing ones, as well as available to people at a construction site without higher mathematical education, which makes it a priority for research. The applicability of this method must be proven by linking empirical coefficients and variables to the conditions of the experiments, while obtaining reliable data that will turn out to be more accurate than the existing calculation methods. This article demonstrates a systematic approach to establishing the suitability of using the method of group analysis of differential equations for problems of winter concreting on the basis of laboratory experiments under stationary conditions. The equations were subject to verification, which, according to the physical description, correspond to the real conditions of the course of thermal processes inside monolithic structures. Based on the obtained processing results, it was decided that it was necessary to further study the innovative method in the conditions of the construction site, but only for some expressions that showed the best results at the stage of laboratory tests.

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Borgese, G., S. Vena, P. Pantano, C. Pace, and E. Bilotta. "Simulation, Modeling, and Analysis of Soliton Waves Interaction and Propagation in CNN Transmission Lines for Innovative Data Communication and Processing." Discrete Dynamics in Nature and Society 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/139238.

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We present an innovative approach to study the interaction between oblique solitons, using nonlinear transmission lines, based on Cellular Neural Network (CNN) paradigm. A single transmission line consists of a 1D array of cells that interact with neighboring cells, through both linear and nonlinear connections. Each cell is controlled by a nonlinear Ordinary Differential Equation, in particular the Korteweg de Vries equation, which defines the cell status and behavior. Two typologies of CNN transmission lines are modelled: crisscross and ring lines. In order to solve KdV equations two different methods are used: 4th-order Runge-Kutta and Forward Euler methods. This is done to evaluate their accuracy and stability with the purpose of implementing CNN transmission lines on embedded systems such as FPGA and microcontrollers. Simulation/analysis Graphic User Interface platforms are designed to conduct numerical simulations and to display elaboration results. From this analysis it is possible both to identify the presence and the propagation of soliton waves on the transmission lines and to highlight the interaction between solitons and rich nonlinear dynamics. With this approach it is possible to simulate and develop the transmission and processing of information within large brain networks and high density sensor systems.

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Zaręba, Mateusz, and Tomasz Danek. "Nonlinear anisotropic diffusion techniques for seismic signal enhancing - Carpathian Foredeep study." E3S Web of Conferences 66 (2018): 01016. http://dx.doi.org/10.1051/e3sconf/20186601016.

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The use of nonlinear anisotropic diffusion algorithm for advanced seismic signal processing in the complicated geological region of Carpathian Foredeep was examined. This technique allows for an improvement of seismic data quality and for more accurate interpretation by the recovery of a significant amount of structural information in the form of a correlating seismic reflections and by preserving true DHI indicators. It also allows searching for more subtle geological structures. Anisotropic diffusion is an iterative image processing algorithm that removes noise by modifying the data by solving partial differential equations. Moreover, it can reduce image noise without blurring the edges between regions of different chrominance or brightness. This filter preserves edges, lines, or other features relevant to the seismic structural and stratigraphic interpretation. The algorithm also enables noise reduction without removing significant information from a seismic section even for high dips values. For a better estimation of anisotropic diffusion structure tensor, the parameterization is done using the depth field and the calculations in the two-way travel time field. The presented research shows the results of using an anisotropic diffusion algorithm for post-stack and migration processing of seismic 3D data collected in Carpathian reservoir rocks of southern Poland.

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Dyvak, M., V. Manzhula, A. Melnyk, and V. Tymchyshyn. "A System for Monitoring Air Pollution by Motor Vehicles Based on an Autonomous Air-Mobile Measuring Complex." Optoelectronic Information-Power Technologies 42, no. 2 (October 26, 2022): 73–83. http://dx.doi.org/10.31649/1681-7893-2021-42-2-73-83.

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The article proposes an approach to constructing a system of complex and uninterrupted monitoring of harmful emissions of motor vehicles into the air. The architecture of the environmental monitoring system for measuring and forecasting the distribution of pollutant concentrations in motor vehicle exhaust gases, among which mainly CO, SO₂, NO₂, and СО₂, is presented. The mobile information and measurement complex Sniffer4D Hyper-local Air Quality Analyzer and a charging station based on solar batteries are used as the hardware. For modeling and forecasting the distribution of concentrations of harmful emissions, mathematical models of the dynamics of the distribution of concentrations of pollutants due to harmful emissions in the exhaust gases of motor vehicles are proposed in the form of differential equations that are analogs of differential equations in partial derivatives, as models of turbulent diffusion and interval models of the distribution of the background level of pollution concentration in the form of nonlinear algebraic equations. Implemented software for data collection, processing (model learning and prediction), and visualization.

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Dyvak, M., V. Manzhula, A. Melnyk, and V. Tymchyshyn. "A System for Monitoring Air Pollution by Motor Vehicles Based on an Autonomous Air-Mobile Measuring Complex." Optoelectronic Information-Power Technologies 42, no. 2 (October 26, 2022): 73–83. http://dx.doi.org/10.31649/1681-7893-2021-41-1-73-83.

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The article proposes an approach to constructing a system of complex and uninterrupted monitoring of harmful emissions of motor vehicles into the air. The architecture of the environmental monitoring system for measuring and forecasting the distribution of pollutant concentrations in motor vehicle exhaust gases, among which mainly CO, SO₂, NO₂, and СО₂, is presented. The mobile information and measurement complex Sniffer4D Hyper-local Air Quality Analyzer and a charging station based on solar batteries are used as the hardware. For modeling and forecasting the distribution of concentrations of harmful emissions, mathematical models of the dynamics of the distribution of concentrations of pollutants due to harmful emissions in the exhaust gases of motor vehicles are proposed in the form of differential equations that are analogs of differential equations in partial derivatives, as models of turbulent diffusion and interval models of the distribution of the background level of pollution concentration in the form of nonlinear algebraic equations. Implemented software for data collection, processing (model learning and prediction), and visualization.

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Gandham, Rajesh, David Medina, and Timothy Warburton. "GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations." Communications in Computational Physics 18, no. 1 (July 2015): 37–64. http://dx.doi.org/10.4208/cicp.070114.271114a.

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AbstractWe discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforthscheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. Furthermore, we developed a unified multi-threading model OCCA. The kernels expressed in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA, and OpenMP. We compare the performance of the OCCA kernels when cross-compiled with these models.

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Bhatti, Muhammad Mubashir, Anwar Shahid, Tehseen Abbas, Sultan Z. Alamri, and Rahmat Ellahi. "Study of Activation Energy on the Movement of Gyrotactic Microorganism in a Magnetized Nanofluids Past a Porous Plate." Processes 8, no. 3 (March 11, 2020): 328. http://dx.doi.org/10.3390/pr8030328.

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The present study deals with the swimming of gyrotactic microorganisms in a nanofluid past a stretched surface. The combined effects of magnetohydrodynamics and porosity are taken into account. The mathematical modeling is based on momentum, energy, nanoparticle concentration, and microorganisms’ equation. A new computational technique, namely successive local linearization method (SLLM), is used to solve nonlinear coupled differential equations. The SLLM algorithm is smooth to establish and employ because this method is based on a simple univariate linearization of nonlinear functions. The numerical efficiency of SLLM is much powerful as it develops a series of equations which can be subsequently solved by reutilizing the data from the solution of one equation in the next one. The convergence was improved through relaxation parameters in the study. The accuracy of SLLM was assured through known methods and convergence analysis. A comparison of the proposed method with the existing literature has also been made and found an excellent agreement. It is worth mentioning that the successive local linearization method was found to be very stable and flexible for resolving the issues of nonlinear magnetic materials processing transport phenomena.

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Xu, Qingzhen. "A Novel Machine Learning Strategy Based on Two-Dimensional Numerical Models in Financial Engineering." Mathematical Problems in Engineering 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/659809.

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Machine learning is the most commonly used technique to address larger and more complex tasks by analyzing the most relevant information already present in databases. In order to better predict the future trend of the index, this paper proposes a two-dimensional numerical model for machine learning to simulate major U.S. stock market index and uses a nonlinear implicit finite-difference method to find numerical solutions of the two-dimensional simulation model. The proposed machine learning method uses partial differential equations to predict the stock market and can be extensively used to accelerate large-scale data processing on the history database. The experimental results show that the proposed algorithm reduces the prediction error and improves forecasting precision.

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Wu, Jianhong, Hossein Zivari-Piran, John D. Hunter, and John G. Milton. "Projective Clustering Using Neural Networks with Adaptive Delay and Signal Transmission Loss." Neural Computation 23, no. 6 (June 2011): 1568–604. http://dx.doi.org/10.1162/neco_a_00124.

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We develop a new neural network architecture for projective clustering of data sets that incorporates adaptive transmission delays and signal transmission information loss. The resultant selective output signaling mechanism does not require the addition of multiple hidden layers but instead is based on the assumption that the signal transmission velocity between input processing neurons and clustering neurons is proportional to the similarity between the input pattern and the feature vector (the top-down weights) of the clustering neuron. The mathematical model governing the evolution of the signal transmission delay, the short-term memory traces, and the long-term memory traces represents a new class of large-scale delay differential equations where the evolution of the delay is described by a nonlinear differential equation involving the similarity measure already noted. We give a complete description of the computational performance of the network for a wide range of parameter values.

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Ismailov, Bakhtiyar, Zhanat Umarova, Khairulla Ismailov, Aibarsha Dosmakanbetova, and Saule Meldebekova. "Mathematical modeling and algorithm for calculation of thermocatalytic process of producing nanomaterial." Indonesian Journal of Electrical Engineering and Computer Science 23, no. 3 (September 1, 2021): 1590. http://dx.doi.org/10.11591/ijeecs.v23.i3.pp1590-1601.

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<p>At present, when constructing a mathematical description of the pyrolysis reactor, partial differential equations for the components of the gas phase and the catalyst phase are used. In the well-known works on modeling pyrolysis, the obtained models are applicable only for a narrow range of changes in the process parameters, the geometric dimensions are considered constant. The article poses the task of creating a complex mathematical model with additional terms, taking into account nonlinear effects, where the geometric dimensions of the apparatus and operating characteristics vary over a wide range. An analytical method has been developed for the implementation of a mathematical model of catalytic pyrolysis of methane for the production of nanomaterials in a continuous mode. The differential equation for gaseous components with initial and boundary conditions of the third type is reduced to a dimensionless form with a small value of the peclet criterion with a form factor. It is shown that the laplace transform method is mainly suitable for this case, which is applicable both for differential equations for solid-phase components and calculation in a periodic mode. The adequacy of the model results with the known experimental data is checked.</p>

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Abid, Salah Hamza, Ahmed J. Obaid, and Jaafer Hmood Eidi. "3^rd International Conference on Mathematics and Applied Sciences (ICMAS), College of Education, Mustansiriyah University, Baghdad, Iraq in 23-24 March, 2022." Journal of Physics: Conference Series 2322, no. 1 (August 1, 2022): 011001. http://dx.doi.org/10.1088/1742-6596/2322/1/011001.

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ICMAS Conference Proceeding is the 3rd Edition of the College of Education, Mustansiriyah University Conferences. ICMAS involved Our Peer-Reviewed Papers that were reviewed and presented through the 3rd International Conference on Mathematics and Applied Sciences (ICMAS) held in the College of Education, Mustansiriyah University, Baghdad, Iraq, from 23 to 24 March 2022. ICMAS Aims to bring together all aspects and research works in a scientific platform to discuss the latest and new trends in topics covered under broad areas such as (but not limited to): Functional, Nodal, and Mathematical Analysis, Dynamic Systems, and Control Systems, Statistics and Operations Research, Numerical Analysis, Differential Equations and Optimization, Algebra, Topological Aspects, Plasma, and Theoretical Physics, Materials Physics, Materials Analysis, Characterization, Optics, Quantum Optics and Lasers, Mathematical Physics, Applied Physics, Statistical physics, Nonlinear systems, Networking, Communications, Image Processing, Data Security, Vacuum Science, Information Technology and Its Applications. The conference is co-published by Journal of Physics: Conference Series, ISSN: 1742-6588, 1742-6596. List of Conference Committees, Organizing Committee, Scientific Committee, International Committee, Conference Tracks, Invited Speakers, Organizing Institutions, Conference Proceeding of 1st Day (23 March 2022), Conference Sessions (Day 1: 23 March 2022), Conference Sessions (Day 2: 24 March 2022), Wishing to Participate with our Next Conference are available in the pdf

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Pesenson, Meyer, William Roby, and Bruce McCollum. "Multiscale Astronomical Image Processing Based on Nonlinear Partial Differential Equations." Astrophysical Journal 683, no. 1 (August 10, 2008): 566–76. http://dx.doi.org/10.1086/589276.

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Shen, Liling. "Parallel Solving Method for the Variable Coefficient Nonlinear Equation." International Journal of Circuits, Systems and Signal Processing 16 (January 10, 2022): 264–71. http://dx.doi.org/10.46300/9106.2022.16.32.

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In view of the inaccuracy of traditional methods for solving nonlinear equations with variable coefficients in parallel, a new method for solving nonlinear equations with variable coefficients is proposed. Using the generalized symmetry group, the variable coefficient of the equation is taken as a new variable which is the same as the state of the original actual physical field. Some relations between variable coefficient equations and their solutions are found. This paper analyzes the meaning of linear differential equation and nonlinear differential equation, the difference between linear differential equation and nonlinear differential equation and their role in physics, and the necessity of solving nonlinear differential equation. By solving the nonlinear equation with variable coefficients, it can be seen that the general methods to solve the nonlinear equation include scattering inversion, Backlund transform and traveling wave solution. Based on the existing methods for solving nonlinear equations with variable coefficients, the homogeneous balance method is combined with the improved truncated expansion method, truncated expansion method and function reduction method, and the Hopf Cole transform and trial function are combined respectively. The above three methods are used to solve nonlinear equations with variable coefficients. Based on KdV Painleve principle, a parallel method for solving nonlinear equations with variable coefficients is proposed. Finally, it is proved that the method is accurate and effective for the parallel solution of nonlinear equations with variable coefficients.

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Saeed, Umer. "Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation." International Journal of Wavelets, Multiresolution and Information Processing 17, no. 04 (July 2019): 1950026. http://dx.doi.org/10.1142/s0219691319500267.

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In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

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Gouasnouane, O., N. Moussaid, S. Boujena, and K. Kabli. "A nonlinear fractional partial differential equation for image inpainting." Mathematical Modeling and Computing 9, no. 3 (2022): 536–46. http://dx.doi.org/10.23939/mmc2022.03.536.

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Image inpainting is an important research area in image processing. Its main purpose is to supplement missing or damaged domains of images using information from surrounding areas. This step can be performed by using nonlinear diffusive filters requiring a resolution of partial differential evolution equations. In this paper, we propose a filter defined by a partial differential nonlinear evolution equation with spatial fractional derivatives. Due to this, we were able to improve the performance obtained by known inpainting models based on partial differential equations and extend certain existing results in image processing. The discretization of the fractional partial differential equation of the proposed model is carried out using the shifted Grünwald–Letnikov formula, which allows us to build stable numerical schemes. The comparative analysis shows that the proposed model produces an improved image quality better or comparable to that obtained by various other efficient models known from the literature.

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LIONS, PIERRE-LOUIS. "AXIOMATIC DERIVATION OF IMAGE PROCESSING MODELS." Mathematical Models and Methods in Applied Sciences 04, no. 04 (August 1994): 467–75. http://dx.doi.org/10.1142/s0218202594000261.

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We briefly review the derivation due to Alvarez, Guichard, Morel and the author of mathematical models in Image Processing. We deduce from classical axions in Computer Vision some nonlinear partial differential equations of evolution type that correspond to general multi-scale analysis (scale-space). We also obtain specific nonlinear models that satisfy additional invariances which are relevant for the analysis of images.

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BEHERA, RATIKANTA, and MANI MEHRA. "APPROXIMATE SOLUTION OF MODIFIED CAMASSA–HOLM AND DEGASPERIS–PROCESI EQUATIONS USING WAVELET OPTIMIZED FINITE DIFFERENCE METHOD." International Journal of Wavelets, Multiresolution and Information Processing 11, no. 02 (March 2013): 1350019. http://dx.doi.org/10.1142/s0219691313500197.

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In this paper, we apply wavelet optimized finite difference method to solve modified Camassa–Holm and modified Degasperis–Procesi equations. The method is based on Daubechies wavelet with finite difference method on an arbitrary grid. The wavelet is used at regular intervals to adaptively select the grid points according to the local behaviour of the solution. The purpose of wavelet-based numerical methods for solving linear or nonlinear partial differential equations is to develop adaptive schemes, in order to achieve accuracy and computational efficiency. Since most of physical and scientific phenomena are modeled by nonlinear partial differential equations, but it is difficult to handle nonlinear partial differential equations analytically. So we need approximate solution to solve these type of partial differential equation. Numerical results are presented for approximating modified Camassa–Holm and modified Degasperis–Procesi equations, which demonstrate the advantages of this method.

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Reich, Sebastian. "On an existence and uniqueness theory for nonlinear differential-algebraic equations." Circuits, Systems, and Signal Processing 10, no. 3 (September 1991): 343–59. http://dx.doi.org/10.1007/bf01187550.

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Huang, Pan, Jun Yue, and Mao Lin Wang. "Coupled Partial Differential Equations Method for InSAS Interferogram Filtering." Advanced Materials Research 487 (March 2012): 103–6. http://dx.doi.org/10.4028/www.scientific.net/amr.487.103.

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In this paper, a coupled nonlinear diffusion partial differential equations (PDE) method for Interferometric Synthetic Aperture Sonar(InSAS) interferogram filtering was introduced. Many previous PDE methods in this area usually use Gauss pre-filtering. The choice of variance in Gauss function plays a very important role in the quality of the image obtained. Manually choice of the variance can hardly reach the self-adaptation aim. Using nonlinear diffusion equation to instead Gauss pre-filtering can overcome the disadvantage mentioned above. Numerical experiment results indicate that this coupled PDE method is able to effectively reduce the noise and preserve edge information. And it is important for InSAS real time processing.

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Yang, Junpu. "Computer Data Encryption System Based on Nonlinear Partial Differential Equations." Mobile Information Systems 2022 (August 19, 2022): 1–9. http://dx.doi.org/10.1155/2022/3395019.

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Data encryption is to convert plaintext data into ciphertext through a data encryption algorithm and then transmit the ciphertext. After the recipient receives the ciphertext, the ciphertext is restored to plaintext, which provides protection and technical support for information security. The main purpose of this article is to design a computer data encryption system based on nonlinear partial differential equations. This paper uses the DES encryption algorithm to encrypt data and implements an onion encryption system that encrypts the outer layer of the database and tests and analyzes the encryption efficiency and additional overhead of the database encryption system on a general database to verify the design application prospects of ideas. In addition, the overall scheme of the encryption system, the hardware, and software of the system are designed in detail, the system is debugged, the overall test is tested, and the data encryption and decryption are effective and feasible. The experimental results of this paper show that after the construction of a computer data encryption system based on nonlinear partial differential equations, the overall security of the data is increased by 25%. In addition, after comparison, the security performance of the onion-type data encryption system is higher than that of the MySQL-type data. The performance of the encryption system is 21% lower. It has certain practical value and significance to apply it to the computer data encryption system.

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Frank, Jason, and Sergiy Zhuk. "A detectability criterion and data assimilation for nonlinear differential equations." Nonlinearity 31, no. 11 (October 18, 2018): 5235–57. http://dx.doi.org/10.1088/1361-6544/aaddcb.

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Saeed, Umer. "Haar wavelet operational matrix method for system of fractional nonlinear differential equations." International Journal of Wavelets, Multiresolution and Information Processing 15, no. 05 (August 28, 2017): 1750043. http://dx.doi.org/10.1142/s0219691317500436.

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In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

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CSEREY, GYÖRGY, CSABA REKECZKY, and PÉTER FÖLDESY. "PDE-BASED HISTOGRAM MODIFICATION WITH EMBEDDED MORPHOLOGICAL PROCESSING OF THE LEVEL-SETS." Journal of Circuits, Systems and Computers 12, no. 04 (August 2003): 519–38. http://dx.doi.org/10.1142/s021812660300101x.

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This paper describes parallel histogram modification techniques with embedded morphological preprocessing methods within the CNN-UM framework. The procedure is formulated in terms of nonlinear partial differential equations (PDE) and approximated through finite differences in space, resulting in coupled nonlinear ordinary differential equations (ODE). The I/O mapping of the system (containing both local and global couplings) can be calculated by a complex analogic (analog and logic) algorithm executed on a stored program nonlinear array processor, called the cellular nonlinear network universal machine (CNN-UM3). We describe and illustrate how the implementation of the algorithm results in an adaptive multi-thresholding scheme when histogram modification is combined with embedded morphological processing at a finite (low) number of gray-scale levels. This has obvious advantages if the further processing steps are segmentation and/or recognition. Experimental results processing real-life and echocardiography images are measured on different hardware/software platforms, including a 64 × 64 CNN-UM chip (ACE4k6,17).

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VAN GENNIP, YVES, and CAROLA-BIBIANE SCHÖNLIEB. "Introduction: Big data and partial differential equations." European Journal of Applied Mathematics 28, no. 6 (November 7, 2017): 877–85. http://dx.doi.org/10.1017/s0956792517000304.

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Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

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Gupta, A. K., and S. Saha Ray. "Wavelet Methods for Solving Fractional Order Differential Equations." Mathematical Problems in Engineering 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/140453.

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Fractional calculus is a field of applied mathematics which deals with derivatives and integrals of arbitrary orders. The fractional calculus has gained considerable importance during the past decades mainly due to its application in diverse fields of science and engineering such as viscoelasticity, diffusion of biological population, signal processing, electromagnetism, fluid mechanics, electrochemistry, and many more. In this paper, we review different wavelet methods for solving both linear and nonlinear fractional differential equations. Our goal is to analyze the selected wavelet methods and assess their accuracy and efficiency with regard to solving fractional differential equations. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on various wavelets in order to solve differential equations of arbitrary order.

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Koo, Ja Choon, H. S. Kim, Jae Boong Choi, and Young Jin Kim. "Application of Discrete Hamilton's Equation for Parallel Processing of Impact Problems." Key Engineering Materials 297-300 (November 2005): 716–21. http://dx.doi.org/10.4028/www.scientific.net/kem.297-300.716.

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Application of Hamilton’s theorem is limited to rigid body dynamics problems in spite of its benefit that always yield a set of first order differential equations as a model. From the fundamental formulation procedure, introduction of Hamilton’s principle to continuum problems differs from the traditional continuum modeling methodology that relies upon partial differential field equation. For the analysis of impact problems where highly nonlinear coupled models are norm, massively distributed computation schemes are usually employed and they significantly reduce computational cost and improve accuracy. With the parallel resources in mind, the present work applies Hamiltonian modeling approach to a shock propagation problem in continuous media. The formulated model which is in first order ordinary differential equations is efficiently calculated on a Beowulf based Linux parallel machines.

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Bilgehan, Bülent, and Ali Özyapıcı. "Direct solution of nonlinear differential equations derived from real circuit applications." Analog Integrated Circuits and Signal Processing 101, no. 3 (July 24, 2019): 441–48. http://dx.doi.org/10.1007/s10470-019-01511-0.

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Bar-Sinai, Yohai, Stephan Hoyer, Jason Hickey, and Michael P. Brenner. "Learning data-driven discretizations for partial differential equations." Proceedings of the National Academy of Sciences 116, no. 31 (July 16, 2019): 15344–49. http://dx.doi.org/10.1073/pnas.1814058116.

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The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.

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Choi, Jeongwhan, Hwangyong Choi, Jeehyun Hwang, and Noseong Park. "Graph Neural Controlled Differential Equations for Traffic Forecasting." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 6 (June 28, 2022): 6367–74. http://dx.doi.org/10.1609/aaai.v36i6.20587.

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Traffic forecasting is one of the most popular spatio-temporal tasks in the field of machine learning. A prevalent approach in the field is to combine graph convolutional networks and recurrent neural networks for the spatio-temporal processing. There has been fierce competition and many novel methods have been proposed. In this paper, we present the method of spatio-temporal graph neural controlled differential equation (STG-NCDE). Neural controlled differential equations (NCDEs) are a breakthrough concept for processing sequential data. We extend the concept and design two NCDEs: one for the temporal processing and the other for the spatial processing. After that, we combine them into a single framework. We conduct experiments with 6 benchmark datasets and 20 baselines. STG-NCDE shows the best accuracy in all cases, outperforming all those 20 baselines by non-trivial margins.

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KARAMI, A., H. R. KARIMI, B. MOSHIRI, and P. JABEDAR MARALANI. "WAVELET-BASED ADAPTIVE COLLOCATION METHOD FOR THE RESOLUTION OF NONLINEAR PDEs." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 06 (November 2007): 957–73. http://dx.doi.org/10.1142/s0219691307002154.

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Theoretical modeling of dynamic processes in chemical engineering often implies the numeric solution of one or more partial differential equations. The complexity of such problems is increased when the solutions exhibit sharp moving fronts. An efficient adaptive multiresolution numerical method is described for solving systems of partial differential equations. This method is based on multiresolution analysis and interpolating wavelets, that dynamically adapts the collocation grid so that higher resolution is automatically attributed to domain regions where sharp features are present. Space derivatives were computed in an irregular grid by cubic splines method. The effectiveness of the method is demonstrated with some relevant examples in a chemical engineering context.

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MEHRA, MANI, and B. V. RATHISH KUMAR. "TIME ACCURATE FAST THREE-STEP WAVELET-GALERKIN METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS." International Journal of Wavelets, Multiresolution and Information Processing 04, no. 01 (March 2006): 65–79. http://dx.doi.org/10.1142/s0219691306001099.

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We introduce the concept of three-step wavelet-Galerkin method based on the Taylor series expansion in time. Unlike the Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving nonlinear problems. Numerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of functions which are smooth, except for localized regions, up to any given accuracy are presented. Here numerical experiments deal with advection equation with the spiky solution in one dimension, two dimensions and nonlinear equation with a shock in solution in two dimensions. Numerical results indicate the versatility and effectiveness of the proposed scheme.

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Sahu, P. K., and S. Saha Ray. "A numerical approach for solving nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions." International Journal of Wavelets, Multiresolution and Information Processing 14, no. 05 (August 24, 2016): 1650036. http://dx.doi.org/10.1142/s0219691316500363.

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In this paper, a numerical approximation based on Legendre wavelets has been developed to solve nonlinear fractional Volterra–Fredholm integro-differential equations. Legendre wavelets are generated by dilation and translation of Legendre polynomials. The properties of the Legendre wavelets are presented in the paper. The proposed wavelet method transforms the integral equations to a system of nonlinear algebraic equations and this algebraic system has been solved numerically by Newton’s method. Convergence analysis of the proposed method has been discussed in this paper. Some examples have been illustrated to show the applicability and accuracy of the present method.

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Kunkel, Peter, and Volker Mehrmann. "Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index." Mathematics of Control, Signals, and Systems 20, no. 3 (August 2008): 227–69. http://dx.doi.org/10.1007/s00498-008-0032-1.

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Peuteman, Joan, and Dirk Aeyels. "Exponential Stability of Nonlinear Time-Varying Differential Equations and Partial Averaging." Mathematics of Control, Signals, and Systems (MCSS) 15, no. 1 (March 1, 2002): 42–70. http://dx.doi.org/10.1007/s004980200002.

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Ma, Wen-Xiu, Mohamed R. Ali, and R. Sadat. "Analytical Solutions for Nonlinear Dispersive Physical Model." Complexity 2020 (August 28, 2020): 1–8. http://dx.doi.org/10.1155/2020/3714832.

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Nonlinear evolution equations widely describe phenomena in various fields of science, such as plasma, nuclear physics, chemical reactions, optics, shallow water waves, fluid dynamics, signal processing, and image processing. In the present work, the derivation and analysis of Lie symmetries are presented for the time-fractional Benjamin–Bona–Mahony equation (FBBM) with the Riemann–Liouville derivatives. The time FBBM equation is reduced to a nonlinear fractional ordinary differential equation (NLFODE) using its Lie symmetries. These symmetries are derivations using the prolongation theorem. Applying the subequation method, we then use the integrating factor property to solve the NLFODE to obtain a few travelling wave solutions to the time FBBM.

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Li, Qiang. "Using Nonlinear Diffusion Model to Identify Music Signals." Advances in Mathematical Physics 2021 (October 13, 2021): 1–11. http://dx.doi.org/10.1155/2021/2210953.

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In this paper, combined with the partial differential equation music signal smoothing model, a new music signal recognition model is proposed. Experimental results show that this model has the advantages of the above two models at the same time, which can remove noise and enhance music signals. This paper also studies the music signal recognition method based on the nonlinear diffusion model. By distinguishing the flat area and the boundary area of the music signal, a new diffusion coefficient equation is obtained by combining these two methods, and the corresponding partial differential equation is discretized by the finite difference method with numerical solution. The application of partial differential equations in music signal processing is a relatively new topic. Because it can accurately model the music signal, it solves many complicated problems in music signal processing. Then, we use the group shift Fourier transform (GSFT) to transform this partial differential equation into a linear homogeneous differential equation system, and then use the series to obtain the solution of the linear homogeneous differential equation system, and finally use the group shift inverse Fourier transform to obtain the noise frequency modulation time-dependent solution of the probability density function of the interference signal. This paper attempts to use the mathematical method of stochastic differentiation to solve the key problem of the time-dependent solution of the probability density function of noise interference signals and to study the application of random differentiation theory in radar interference signal processing and music signal processing. At the end of the thesis, the application of stochastic differentiation in the filtering processing of music signals is tried. According to the inherent self-similarity of the music signal system and the completeness and stability of the empirical mode decomposition (EMD) algorithm, a new kind of EMD music using stochastic differentiation is proposed for signal filtering algorithm. This improved anisotropic diffusion method can maintain and enhance the boundary while smoothing the music signal. The filtering results of the actual music signal show that the algorithm is effective.

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Hu, Peng, and Chengming Huang. "Analytical and numerical stability of nonlinear neutral delay integro-differential equations." Journal of the Franklin Institute 348, no. 6 (August 2011): 1082–100. http://dx.doi.org/10.1016/j.jfranklin.2011.04.007.

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Yang, Meng, Jian Zhang, and Xingjiu Luo. "Research on New Types of Suspension Vibration Reduction Systems (SVRSs) with Geometric Nonlinear Damping." Mathematical Problems in Engineering 2021 (April 7, 2021): 1–12. http://dx.doi.org/10.1155/2021/6627693.

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Firstly, this paper puts forward two new types of suspension vibration reduction systems (SVRSs) with geometric nonlinear damping based on general SVRS (GSVRS), which only has geometric nonlinear stiffness. Secondly, it derives the motion differential equations for the two new types of SVRS, respectively, and discusses the similarities and differences among the two types and GSVRS through the comparison of motion differential equations. Then, it conducts dimensionless processing of the motion differential equations for the two new types of SVRS and carries out a comparative study on the vibration isolation performance of the two types of SVRS under impact excitation and random excitation, respectively. At last, it performs the optimal computation of the chosen new type of SVRS through the ergodic optimization method and studies the influence rule of SVRS parameters on vibration isolation performance so as to realize the optimization of vibration isolation performance.

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Yan, Zuomao, and Xiumei Jia. "Optimal Solutions of Fractional Nonlinear Impulsive Neutral Stochastic Functional Integro-Differential Equations." Numerical Functional Analysis and Optimization 40, no. 14 (June 27, 2019): 1593–643. http://dx.doi.org/10.1080/01630563.2018.1501060.

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Ma, Baoshan, Mingkun Fang, and Xiangtian Jiao. "Inference of gene regulatory networks based on nonlinear ordinary differential equations." Bioinformatics 36, no. 19 (October 1, 2020): 4885–93. http://dx.doi.org/10.1093/bioinformatics/btaa032.

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Abstract Motivation Gene regulatory networks (GRNs) capture the regulatory interactions between genes, resulting from the fundamental biological process of transcription and translation. In some cases, the topology of GRNs is not known, and has to be inferred from gene expression data. Most of the existing GRNs reconstruction algorithms are either applied to time-series data or steady-state data. Although time-series data include more information about the system dynamics, steady-state data imply stability of the underlying regulatory networks. Results In this article, we propose a method for inferring GRNs from time-series and steady-state data jointly. We make use of a non-linear ordinary differential equations framework to model dynamic gene regulation and an importance measurement strategy to infer all putative regulatory links efficiently. The proposed method is evaluated extensively on the artificial DREAM4 dataset and two real gene expression datasets of yeast and Escherichia coli. Based on public benchmark datasets, the proposed method outperforms other popular inference algorithms in terms of overall score. By comparing the performance on the datasets with different scales, the results show that our method still keeps good robustness and accuracy at a low computational complexity. Availability and implementation The proposed method is written in the Python language, and is available at: https://github.com/lab319/GRNs_nonlinear_ODEs Supplementary information Supplementary data are available at Bioinformatics online.

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Sobamowo, M. G., A. T. Akinshilo, and A. A. Yinusa. "Thermo-Magneto-Solutal Squeezing Flow of Nanofluid between Two Parallel Disks Embedded in a Porous Medium: Effects of Nanoparticle Geometry, Slip and Temperature Jump Conditions." Modelling and Simulation in Engineering 2018 (June 3, 2018): 1–18. http://dx.doi.org/10.1155/2018/7364634.

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The various applications of squeezing flow between two parallel surfaces such as those that are evident in manufacturing industries, polymer processing, compression, power transmission, lubricating system, food processing, and cooling amongst others call for further study on the effects of various parameters on the flow phenomena. In the present study, effects of nanoparticle geometry, slip, and temperature jump conditions on thermo-magneto-solutal squeezing flow of nanofluid between two parallel disks embedded in a porous medium are investigated, analyzed, and discussed. Similarity variables are used to transform the developed governing systems of nonlinear partial differential equations to systems of nonlinear ordinary differential equations. Homotopy perturbation method is used to solve the systems of the nonlinear ordinary differential equations. In order to verify the accuracy of the developed analytical solutions, the results of the homotopy perturbation method are compared with the results of the numerical method using the shooting method coupled with the fourth-order Runge–Kutta, and good agreements are established. Through the approximate analytical solutions, parametric studies are carried out to investigate the effects of nanoparticle size and shape, Brownian motion parameter, nanoparticle parameter, thermophoresis parameter, Hartmann number, Lewis number and pressure gradient parameters, slip, and temperature jump boundary conditions on thermo-solutal and hydromagnetic behavior of the nanofluid. This study will enhance and advance the understanding of nanofluidics such as energy conservation, friction reduction, and micromixing of biological samples.

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CARDIEL, ROSA E., ELENA I. KAIKINA, and PAVEL I. NAUMKIN. "ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE." Communications in Contemporary Mathematics 08, no. 02 (April 2006): 189–217. http://dx.doi.org/10.1142/s021919970600209x.

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We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line [Formula: see text] where the number M depends on the order of the pseudo-differential operator [Formula: see text] on a half-line. The nonlinear term [Formula: see text] is such that [Formula: see text] as u, v → 0, with ρ, σ > 0. Pseudo-differential operator [Formula: see text] is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions.

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DECK, THOMAS. "CONTINUOUS DEPENDENCE ON INITIAL DATA FOR SOLUTIONS OF NONLINEAR STOCHASTIC EVOLUTION EQUATIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 05, no. 03 (September 2002): 333–50. http://dx.doi.org/10.1142/s0219025702000870.

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We consider stochastic evolution equations in the framework of white noise analysis. Contraction operators on inductive limits of Banach spaces arise naturally in this context and we first extend Banach's fixed point theorem to this type of spaces. In order to apply the fixed point theorem to evolution equations, we construct a topological isomorphism between spaces of generalized random fields and the corresponding spaces of U-functionals. As an application we show that the solutions of some nonlinear stochastic heat equations depend continuously on their initial data. This method also applies to stochastic Volterra equations, stochastic reaction–diffusion equations and to anticipating stochastic differential equations.

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Du, Wenting, and Jin Su. "Uncertainty Quantification for Numerical Solutions of the Nonlinear Partial Differential Equations by Using the Multi-Fidelity Monte Carlo Method." Applied Sciences 12, no. 14 (July 12, 2022): 7045. http://dx.doi.org/10.3390/app12147045.

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The Monte Carlo simulation is a popular statistical method to estimate the effect of uncertainties on the solutions of nonlinear partial differential equations, but it requires a huge computational cost of the deterministic model, and the convergence may become slow. For this reason, we developed the multi-fidelity Monte Carlo (MFMC) methods based on data-driven low-fidelity models for uncertainty analysis of nonlinear partial differential equations. Firstly, the nonlinear partial differential equations are transformed into ordinary differential equations (ODEs) by using finite difference discretization or Fourier transformation. Then, the reduced dimension model and discrete empirical interpolation method (DEIM) are coupled to construct effective nonlinear low-fidelity models in ODEs system. Finally, the MFMC method is used to combine the output information of the high-fidelity model and the low-fidelity models to give the optimal estimation of the statistics. Experimental results of the nonlinear Schrodinger equation and the Burgers’ equation show that, compared with the standard Monte Carlo method, the MFMC method based on the data-driven low-fidelity model in this paper can improve the calculation efficiency significantly.

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KARIMI, HAMID REZA, MAURICIO ZAPATEIRO, and NINGSU LUO. "WAVELET-BASED PARAMETER IDENTIFICATION OF A NONLINEAR MAGNETORHEOLOGICAL DAMPER." International Journal of Wavelets, Multiresolution and Information Processing 07, no. 02 (March 2009): 183–98. http://dx.doi.org/10.1142/s0219691309002842.

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In recent years, significant advances in vibration control of structures have been achieved due greatly to the emergent technologies based on smart materials, such as mangnetorheological (MR) fluids. This paper develops a computational algorithm for the modeling and identification of the MR dampers by using wavelet systems to handle the nonlinear terms. By taking into account the Haar wavelets, the properties of integral operational matrix and of product operational matrix are introduced and utilized to find an algebraic representation form instead of the differential equations of the dynamical system. It is shown that MR damper parameters can be estimated easily by considering only the algebraic equations obtained.

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Xie, Ganquan, Jianhua Li, Ernest L. Majer, Daxin Zuo, and Michael L. Oristaglio. "3-D electromagnetic modeling and nonlinear inversion." GEOPHYSICS 65, no. 3 (May 2000): 804–22. http://dx.doi.org/10.1190/1.1444779.

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We describe a new algorithm for 3-D electromagnetic inversion that uses global integral and local differential equations for both the forward and inverse problems. The coupled integral and differential equations are discretized by the finite element method and solved on a parallel computer using domain decomposition. The structure of the algorithm allows efficient solution of large 3-D inverse problems. Tests on both synthetic and field data show that the algorithm converges reliably and efficiently and gives high‐resolution conductivity images.

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MÜLLER, T. G., and J. TIMMER. "PARAMETER IDENTIFICATION TECHNIQUES FOR PARTIAL DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 14, no. 06 (June 2004): 2053–60. http://dx.doi.org/10.1142/s0218127404010424.

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Many physical systems exhibiting nonlinear spatiotemporal dynamics can be modeled by partial differential equations. Although information about the physical properties for many of these systems is available, normally not all dynamical parameters are known and, therefore, have to be estimated from experimental data. We analyze two prominent approaches to solve this problem and describe advantages and disadvantages of both methods. Specifically, we focus on the dependence of the quality of the parameter estimates with respect to noise and temporal and spatial resolution of the measurements.

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Schaeffer, Hayden. "Learning partial differential equations via data discovery and sparse optimization." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2197 (January 2017): 20160446. http://dx.doi.org/10.1098/rspa.2016.0446.

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We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

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Ahmad, Jamshad, Angbeen Iqbal, and Qazi Mahmood Ul Hassan. "Study of Nonlinear Fuzzy Integro-differential Equations Using Mathematical Methods and Applications." INTERNATIONAL JOURNAL of FUZZY LOGIC and INTELLIGENT SYSTEMS 21, no. 1 (March 31, 2021): 76–85. http://dx.doi.org/10.5391/ijfis.2021.21.1.76.

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Journal articles on the topic 'Differential equations, Nonlinear Data processing'

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