Technique For Differential Equations Research Articles

Tensor Product Technique and Atomic Solution of Fractional Partial Differential Equations

JEL Classification: 34G10, 34A55

Monotone iterative technique for multi-term time fractional measure differential equations

Monotone iterative technique for multi-term time fractional measure differential equations

Deep learning bulk spacetime from boundary optical conductivity

We employ a deep learning method to deduce the bulk spacetime from boundary optical conductivity. We apply the neural ordinary differential equation technique, tailored for continuous functions such as the metric, to the typical class of holographic condensed matter models featuring broken translations: linear-axion models. We successfully extract the bulk metric from the boundary holographic optical conductivity. Furthermore, as an example for real material, we use experimental optical conductivity of UPd2Al3, a representative of heavy fermion metals in strongly correlated electron systems, and construct the corresponding bulk metric. To our knowledge, our work is the first illustration of deep learning bulk spacetime from boundary holographic or experimental conductivity data.

Double diffusive on powell eyring fluid flow by mixed convection from an exponential stretching surface with variable viscosity/thermal conductivity

This study delves into heat and mass transference in fluids with variable thermo-physical features, focusing on the Powell-Eyring fluid, a non-Newtonian substance with unique characteristics. Our aim is to understand the behaviour of this shear-thinning fluid when interacting with an exponentially stretchable surface, considering factors like mixed convection and thermal radiation. Our research endeavours to uncover novel findings in this intricate domain, with the primary objective of comprehending how this shear-thinning fluid behaves when it encounters an exponentially stretchable surface. To address the complexities, we employ the BVP4C technique in MATLAB, transforming governing equations into nonlinear partial differential equations (PDEs) based on mass, linear momentum, and energy conservation principles. The Homotopy method aids in navigating these equations, and numerical solutions are calculated using the powerful BVP4C technique for ordinary differential equations (odes). The research stands out for its comprehensive exploration of the interplay among diverse factors, including radiation, variable viscosity, mixed convection, and activation energy constraints. Findings are presented through tables and graphs, offering valuable insights into the intricate physical phenomena within this multifaceted field.

Differential Fault and Algebraic Equation Combined Analysis on PICO

In modern information technology, research on block cipher security is imperative. Concerning the ultra lightweight block cipher PICO, there has been only one study focused on recovering its complete master key, with a large search space of 264, and no fault analysis yet. This paper proposes a new fault analysis approach, combining differential fault and algebraic equation techniques. It achieved the recovery of PICO’s entire master key with 40 faults in an average time of 0.57 h. S-box decomposition was utilized to optimize our approach, reducing the time by a remarkable 75.83% under the identical 40-fault condition. Furthermore, PICO’s complete master key could be recovered with 28 faults in an average time of 0.78 h, indicating a significant 237 reduction in its search space compared to the previous study. This marks the first fault analysis on PICO. Compared to conventional fault analysis methods DFA (differential fault analysis) and AFA (algebraic fault analysis), our approach outperforms in recovering PICO’s entire master key, highlighting the cruciality of key expansion complexity in block cipher security. Therefore, our approach could serve to recover master keys of block ciphers with comparably complicated key expansions, and production of more secure block ciphers could result.

Portfolio-consumption choice problem with voluntary retirement and consumption constraints

We study the investment–consumption choice problem with voluntary retirement and upside/downside consumption constraints in infinite horizon, which can be formulated as a two-phase mixed stochastic control problem including stopping time coupled with controls. By verification theorem, we show the strong solution of a fully nonlinear differential variational inequality, satisfying some special properties, is the value function of this problem. Then we eliminate the nonlinearity through Legendre transformation, and obtain the closed-form formulas of the optimal investment–consumption feedback strategy, the optimal retirement strategy and the value function by solving the dual equation. Since the introduction of consumption constraints, these formulas are both segmented and semi-explicit, and bring certain difficulties to the analysis of the properties of the optimal strategies and the value function. The regularity of the feedback strategy and the value function are obtained, as well as their monotonicity or non-monotonicity with respect to various parameters are discussed by some stochastic analysis methods and differential equation techniques.

Accuracy Measurement of Hyperspectral Image Classification in Remote Sensing with the Light Spectrum-based Affinity Propagation Clustering-based Segmentation

The area of remote sensing and computer vision includes the challenge of hyperspectral image classification. It entails grouping pixels in hyperspectral pictures into several classes according to their spectral signature. Hyperspectral photographs are helpful for a variety of applications, including vegetation study, mineral mapping, and mapping urban land use, since they include information on an object's reflectance in hundreds of small, contiguous wavelength bands. This task's objective is to correctly identify and categorize several item categories in the image. Many approaches have been stated by several researchers in this field to enhance the accuracy of the segmentation and accuracy. However, fails to attain the optimal accuracy due to the intricate nature of the images. To tackle these issues, we propose a novel Modified Extreme Learning machine (M-ELM) approach for the credible hyperspectral image classification outcomes with the publicly available Kaggle datasets. Before the classification, the input images are segmented using the Light Spectrum-based modified affinity propagation clustering technique (LSO-MAPC). In the beginning, the images are pre-processed using the non-linear diffusion partial differential equations technique which effectively pre-processed the image spatially. Experiments are effectuated to analyze the performance of the proposed method and compared it with state-of-art works in a quantitative way. The proposed approach ensures a classification accuracy of 96%.

Model reduction for stochastic systems with nonlinear drift

In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subspace is identified leading to a balancing related highly accurate reduced order SDE. We provide an algebraic error criterion and an error analysis of the propose model reduction schemes. The paper is concluded by applying our method to spatially discretized reaction diffusion equations.

The relationship between two kinds of structural credit migration models

<abstract><p>In this paper, we presented an asymptotic relationship between the single threshold model and the model with different upgrade and downgrade thresholds for credit migration problems with fixed boundaries. By partial differential equation (PDE) techniques, we proved that the solution of the asymmetric threshold problem converges to that of a single threshold problem uniformly as one of the asymmetric thresholds approaches the other.</p></abstract>

Time-domain modeling of interband transitions in plasmonic systems

Efficient modeling of dispersive materials via time-domain simulations of the Maxwell equations relies on the technique of auxiliary differential equations. In this approach, a material’s frequency-dependent permittivity is represented via a sum of rational functions, e.g., Lorentz poles, and the associated free parameters are determined by fitting to experimental data. In the present work, we present a modified approach for plasmonic materials that requires considerably fewer fit parameters than traditional approaches. Specifically, we consider the underlying microscopic theory and, in the frequency domain, separate the hydrodynamic contributions of the quasi-free electrons in partially filled bands from the interband transitions. As an illustration, we apply our approach to gold and demonstrate how to treat the interband transitions within the effective model via connecting to the underlying electronic band structure, thereby assigning physical meaning to the remaining fit parameters. Finally, we show how to utilize this approach within the technique of auxiliary differential equations. Our approach can be extended to other plasmonic materials and leads to efficient time-domain simulations of plasmonic structures for frequency ranges where interband transitions have to be considered.

Similarity solution of fractional viscoelastic magnetohydrodynamic fluid flow over a permeable stretching sheet with suction/injection

A novel easy‐to‐use similarity technique for spatial fractional partial differential equations is introduced to solve the flow and heat transfer of fractional viscoelastic magnetohydrodynamic (MHD) fluid over a permeable stretching sheet. The proposed method represents a new technique for deriving similarity transformations, which allows the conversion of the fractional partial differential equations to fractional ordinary differential equations. Moreover, using the Grünwald–Letnikov formula, the fourth‐order Runge–Kutta method is adopted here to solve the obtained similarity equations, using the shooting method. The transformed governing equations were found to be dependent on six physical parameters, namely, fractional parameter, magnetic field parameter, generalized local Reynolds number, generalized Prandtl number, the wall suction/injection parameter, and the wall stretching exponential. The results reveal that maximums of the skin friction coefficient and local Nusselt number are obtained for fractional‐order derivative of about 0.5. Moreover, application of a magnetic field on electrically conducting fractional fluid thins the velocity boundary layer and thickens the thermal boundary layer; therefore, the skin friction coefficient increases whereas the local Nusselt number decreases. Furthermore, an increase in any parameters of the local Reynolds number, suction parameter, or the wall stretching exponential leads to thinner boundary layers and higher values of skin friction coefficient and local Nusselt number, whereas imposing injection has opposite effects. Finally, increasing the generalized Prandtl number leads to a thinner thermal boundary layer and higher local Nusselt number for the fractional viscoelastic fluid. It would be of interest to extend the proposed method to other practical problems in the same field.

RETRACTED: Analysis of governing equations of multibody dynamic systems through kinematic properties of joints and effects of kinematic chains on closed-form mechanism

In this investigation, the fundamental equations dictating the behavior of dynamic systems were ascertained by means of an innovative approach that amalgamated the kinematic properties of joints and the intricate kinematic chains of multibody systems into a series of governing equations. Subsequently, the equations that govern the behavior of multibody systems were converted into ordinary differential equations (ODEs) utilizing the calculus of matrix-valued functions. This computational technique is adept at swiftly acquiring recursive differential equations of motion for multibody systems. Thus, the expenditure of computational costs associated with the simulation was efficaciously minimized. The mechanisms of Andrew's squeezing and carpet scrapping were employed in conjunction with kinematic constraints to authenticate the validity of the proposed methodology. The findings suggest that, notwithstanding the fact that the root mean square error (RMSE) in the generalized coordinates between the two ODE and differential algebraic equation (DAE) techniques is small in the Andrew's squeezing mechanism (at approximately 0.02) and in the carpet scraping mechanism (at approximately 0.7), there are still discrepancies between the two methods. The degree of convergence in the aforementioned methodologies, despite the slight RMSE difference between them, shows a clear disparity. By way of illustration, when employing the conventional DAE technique, the duration required to resolve the Andrew's squeezing and carpet scraping mechanisms is 5 and 11.34 s, respectively. However, by utilizing the ODE methodology in the aforementioned mechanisms, the time required to determine a solution is truncated to 1.19 and 2.1 s, sequentially.

A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

Similarity Transforming Techniques of Partial Differential Equations and its Applications: A review

Similarity Transforming Techniques of Partial Differential Equations and its Applications: A review

Accounting for unobserved spatial variation in step selection analyses of animal movement via spatial random effects

Abstract Step selection analysis (SSA) is a common framework for understanding animal movement and resource selection using telemetry data. Such data are, however, inherently autocorrelated in space, a complication that could impact SSA‐based inference if left unaddressed. Accounting for spatial correlation is standard statistical practice when analysing spatial data, and its importance is increasingly recognized in ecological models (e.g. species distribution models). Nonetheless, no framework yet exists to account for such correlation when analysing animal movement using SSA. Here, we extend the popular method integrated step selection analysis (iSSA) by including a Gaussian field (GF) in the linear predictor to account for spatial correlation. For this, we use the Bayesian framework R‐INLA and the stochastic partial differential equations (SPDE) technique. We show through a simulation study that our method provides accurate fixed effects estimates, quantifies their uncertainty well and improves the predictions. In addition, we demonstrate the practical utility of our method by applying it to three wolverine (Gulo gulo) tracks. Our method solves the problems of assuming spatially independent residuals in the SSA framework. In addition, it offers new possibilities for making long‐term predictions of habitat usage.

Data-Completion and Model Correction by Means of Evanescent Regularization

System components are often regarded as part of a whole system, especially when it comes to data-driven modeling. Thus, subsystem modeling is disregarded in general when building a data-driven response, especially since multiple subsystem outputs are never measured in real applications. However, subsystem knowledge and accurate modeling are of utmost importance when aiming to repair, tune or troubleshoot a system. This work proposes a holistic modeling of subsystems in an embedded system setting. A hybrid modeling starting from the physics-based model is proposed in this work, correcting or enhancing the model, and predicting output variables, even when a measurement is never available for some of those variables. The process relies on the variables’ history, and employs an adjoint-free neural ordinary differential equation technique, along with evanescent regularization to enhance the convergence on the unmeasurable variables. The updated model converges to the exact measurements, for both the measurable and the unmeasurable variables. Multiple examples are presented using synthetic data, to allow an easy evaluation of the hidden or unmeasurable variables. The relative error offered by the updated model is around 0.001% for the measurable quantities and 0.1% for the unmeasurable ones.

Numerical approximation with the splitting algorithm to a solution of the modified regularized long wave equation

In this article, a Lie-Totter splitting algorithm, which is highly reliable, flexible and convenient, is proposed along with the collocation finite element method to approximate solutions of the modified regular long wave equation. For this article, quintic B-spline approximation functions are used in the implementation of collocation methods. Four numerical examples including a single solitary wave, the interaction of two- three solitary waves, and a Maxwellian initial condition are presented to test the closeness of the solutions obtained by the proposed algorithm to the exact solutions. The solutions produced are compared with those in some studies with the same parameters that exist in the literature. The fact that the present algorithm produces results as intended is a proof of how useful, accurate and reliable it is. It can be stated that this fact will be very useful the application of the presented technique for other partial differential equations, with the thought that it may lead the reader to obtain superior results from this study.

Stochastic modeling for describing crystallization droplets in water emulsion

This paper introduces a new, stochastic mathematical model for the crystallization of emulsion in dispersed media. The mathematical model reads as a stochastic partial differential equation by combining the heat energy equation and the nucleation theory with specified drift and diffusion. We show the existence and uniqueness of the solution of the model by using techniques of stochastic partial differential equations. Numerical experiments are drawn to support the theoretical results. Moreover, comparison of numerical results to experimental ones is provided.

Analysis of electromagnetic/thermal coupling of Debye media using HIE-FDTD method

In this paper, the multi-physical working mechanism of the Debye dispersive media is studied in detail, and the coupling analysis of Maxwell's and heat conduction equations is carried out. The hybrid-implicit-explicit finite difference time domain (HIE-FDTD) method is applied to accelerate calculation of the electromagnetic(EM). The Debye model, introduced to HIE-FDTD through the auxiliary difference equation (ADE) technique by utilizing polarization current, describes the dispersion characteristics of a water-based absorber. In addition, a new expression of transient power loss density is proposed with the polarization current, which overcomes the negative power loss in the traditional method and makes the computation more accurate. Finally, the analysis proves that the ADE-HIE-FDTD method has good performance in the EM-thermal coupling calculation, and the absorption rate of the absorber shows the interaction effect between temperature distribution and dielectric material.

Fractional Abel Differential Equation Application in Edge Detection: FADEED

There is a long history of perturbed implementations of Abel differential equations in dynamics, linear systems with stochasticity, modeling approaches, and linear algebra. Numerous research has been undertaken, the bulk of which have focused on Abel differential equation techniques and the use of the Abel differential equation using the variation iteration method. Edges define boundaries and are hence of primary relevance in image processing. Edge detection removes unnecessary data, noise, and frequencies from a picture while maintaining crucial structural aspects. To extract edges based on canny, the suggested technique employs fractional Abel differential equation (FADE) logic. In this case, a picture is used with the windowing technique and is submitted to a series of Abel differential equations. The purpose of this research is to assess the performance of a FADE system in edge detection. The results produced by the linear canny operator are compared to those obtained with pictures with substantial contrast variation. This experimental investigation will provide a faster procedure and a better output image than the current methods. The main goal of the proposed technique is to use the edge points and the information they collect for edge identification, resulting in faster and more accurate results. For all of the experimental experiments, MATLAB software was employed.