Form Of Partial Differential Equations Research Articles

Symmetry considerations for differential equation formulations from classical and fractional Lagrangians

The utility of Noether’s classical theorem on differential equations extended to a generalized nonclassical theorem is the focus of this paper. After addressing a couple of standard related Partial Differential Equation (P.D.E.) formulations from classical Lagrangians, it culminates into a non-classical formulation of the diffusion equation in one spatial dimension from a fractional Lagrangian. Comparisons and contrasts between techniques for the classical and fractional formulations, as done here, facilitate the basic computational methods required for building analytical results. A noteworthy interface between Distribution theory, Trace theory and Lie symmetry theory is a key point of interest in this study.

Liouville partial-differential-equation methods for computing 2D complex multivalued eikonals in attenuating media

We have developed a Liouville partial-differential-equation (PDE)-based method for computing complex-valued eikonals in real phase space in the multivalued sense in attenuating media with frequency-independent qualify factors, where the new method computes the real and imaginary parts of the complex-valued eikonal in two steps by solving Liouville equations in real phase space. Because the earth is composed of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is one essential ingredient for describing wave propagation in attenuating media because this unique quantity summarizes two wave properties into one function: Its real part describes the wave kinematics and its imaginary part captures the effects of phase dispersion and amplitude attenuation. Because some popular ordinary-differential-equation (ODE)-based ray-tracing methods for computing complex-valued eikonals in real space distribute the eikonal function irregularly in real space, we are motivated to develop PDE-based Eulerian methods for computing such complex-valued eikonals in real space on regular meshes. Therefore, we solved novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We call the resulting method the Liouville PDE method for complex-valued multivalued eikonals in attenuating media; moreover, this new method provides a unified framework for Eulerianizing several popular approximate real-space ray-tracing methods for complex-valued eikonals, such as viscoacoustic ray tracing, real viscoelastic ray tracing, and real elastic ray tracing. In addition, we also provide Liouville PDE formulations for computing multivalued ray amplitudes in a weakly viscoacoustic medium. Numerical examples, including a synthetic gas-cloud model, demonstrate that our methods yield highly accurate complex-valued eikonals in the multivalued sense.

Mathematical Modeling of Lymph Node Drainage Function by Neural Network

The lymph node (LN) represents a key structural component of the lymphatic system network responsible for the fluid balance in tissues and the immune system functioning. Playing an important role in providing the immune defense of the host organism, LNs can also contribute to the progression of pathological processes, e.g., the spreading of cancer cells. To gain a deeper understanding of the transport function of LNs, experimental approaches are used. Mathematical modeling of the fluid transport through the LN represents a complementary tool for studying the LN functioning under broadly varying physiological conditions. We developed an artificial neural network (NN) model to describe the lymph node drainage function. The NN model predicts the flow characteristics through the LN, including the exchange with the blood vascular systems in relation to the boundary and lymphodynamic conditions, such as the afferent lymph flow, Darcy’s law constants and Starling’s equation parameters. The model is formulated as a feedforward NN with one hidden layer. The NN complements the computational physics-based model of a stationary fluid flow through the LN and the fluid transport across the blood vessel system of the LN. The physical model is specified as a system of boundary integral equations (IEs) equivalent to the original partial differential equations (PDEs; Darcy’s Law and Starling’s equation) formulations. The IE model has been used to generate the training dataset for identifying the NN model architecture and parameters. The computation of the output LN drainage function characteristics (the fluid flow parameters and the exchange with blood) with the trained NN model required about 1000-fold less central processing unit (CPU) time than computationally tracing the flow characteristics of interest with the physics-based IE model. The use of the presented computational models will allow for a more realistic description and prediction of the immune cell circulation, cytokine distribution and drug pharmacokinetics in humans under various health and disease states as well as assisting in the development of artificial LN-on-a-chip technologies.

Computational Modeling of Hybrid Sisko Nanofluid Flow over a Porous Radially Heated Shrinking/Stretching Disc

The present study reveals the behavior of shear-thickening and shear-thinning fluids in magnetohydrodynamic flow comprising the significant impact of a hybrid nanofluid over a porous radially shrinking/stretching disc. The features of physical properties of water-based Ag/TiO2 hybrid nanofluid are examined. The leading flow problem is formulated initially in the requisite form of PDEs (partial differential equations) and then altered into a system of dimensionless ODEs (ordinary differential equations) by employing suitable variables. The renovated dimensionless ODEs are numerically resolved using the package of boundary value problem of fourth-order (bvp4c) available in the MATLAB software. The non-uniqueness of the results for the various pertaining parameters is discussed. There is a significant enhancement in the rate of heat transfer, approximately 13.2%, when the impact of suction governs about 10% in the boundary layer. Therefore, the heat transport rate and the thermal conductivity are greater for the new type of hybrid nanofluid compared with ordinary fluid. The bifurcation of the solutions takes place in the problem only for the shrinking case. Moreover, the sketches show that the nanoparticle volume fractions and the magnetic field delay the separation of the boundarylayer.

Goal-oriented model reduction for parametrized time-dependent nonlinear partial differential equations

We present a projection-based model reduction formulation for parametrized time-dependent nonlinear partial differential equations (PDEs). Our approach builds on the following ingredients: reduced bases (RB), which provide rapidly convergent approximations of the parameter-temporal solution manifold; reduced quadrature (RQ) rules, which provide hyperreduction of the nonlinear residual; and the dual-weighted residual (DWR) method, which provides an error representation formula for the quantity of interest. To find the RQ rules, we develop an empirical quadrature procedure (EQP) for time-dependent problems; we analyze the output error due to hyperreduction using a space–time DWR framework and identify appropriate constraints so that the output error due to hyperreduction is controlled. We in addition equip our reduced model with an online-efficient DWR a posteriori error estimate for the output; we again analyze the error in the hyperreduced dual problem and DWR expression to find appropriate constraints for the EQP so that the error in the error estimate is controlled. In the offline stage, the RBs and RQs, as well as the finite element mesh, are simultaneously constructed using a POD-greedy algorithm that leverages the online-efficient output error estimate. We demonstrate the framework for parametrized unsteady flows in a lid-driven cavity and over a NACA0012 airfoil. Reduced models achieve over two orders of magnitude reduction in the number of degrees of freedom, number of quadrature points, and wall-clock computational time, while achieving less than 0.5% output error and providing efficient error estimates in predictive settings.

Eulerian algorithms for computing some Lagrangian flow network quantities

Originated from the complex network theory, the Lagrangian flow network (LFN) has motivated various tools to analyze fluid flows' transport and mixing behaviors. These tools include the so-called finite time escape rate (FTER), the transition matrix, and also the finite time entropy. This paper proposes novel Eulerian algorithms to compute these LFN quantities in two dimensions based on a partial differential equation (PDE) formulation for the underlying flow map. Compared to the typical Lagrangian computations based on ray tracing, our numerical approaches are computationally efficient. Moreover, for velocity fields obtained numerically by computational fluid dynamic (CFD) solvers, our approach does not require further interpolation of the velocity field but can use the discrete velocity data directly. Finally, we will apply our proposed algorithms to several test examples, including a real-life dataset, to demonstrate the performance of the method.

Dynamic Selective Edge-Based Integer/Fractional-Order Partial Differential Equation for Degraded Document Image Binarization

Conventional thresholding algorithms have had limited success with degraded document images. Recently, partial differential equations (PDEs) have been applied with good results. However, these are usually tailored to handle relatively few specific distortions. In this study, we combine an edge detection term with a linear binarization source term in a PDE formulation. Additionally, a new proposed diffusivity function further amplifies desired edges. It also suppresses undesired edges that comprise bleed-through effects. Furthermore, we develop the fractional variant of the proposed scheme, which further improves results and provides more flexibility. Moreover, nonlinear color spaces are utilized to improve binarization results for images with color distortion. The proposed scheme removes document image degradation such as bleed-through, stains, smudges, etc., and also restores faded text in the images. Experimental subjective and objective results show consistently superior performance of the proposed approach compared to the state-of-the-art PDE-based models.

Optimal system, invariance analysis of fourth-Order nonlinear ablowitz-Kaup-Newell-Segur water wave dynamical equation using lie symmetry approach

Nonlinear science is very common and important natural phenomenon in our surroundings. It is quite possible to find a large number of phenomena, which become a cause of the formulation of a non-linear partial differential equation. Due to the presence of non-linear partial differential equations in each branch of science, these equations have become a useful tool to deal with complex natural phenomena. It is interesting to investigate any complex non-linear partial differential equations for different exact solutions and examine the behavior of the solutions. Many effective approaches are developed to obtain the explicit exact solutions of the NLPDEs. Lie symmetry analysis is also one of the significant approaches to investigate the NLPDEs. Based on the Lie group analysis, we investigate a very famous and important equation, which is named as fourth-order Ablowitz-Kaup-Newell-Segur water wave dynamical equation. The symmetry groups, Commutator Tables and Adjoint of infinitesimals are constructed for this equation. Further, using the adjoint table, the optimal system is obtained. According to the optimal system, we tried to find the possible exact solution using symmetry reduction and presented a brief study of the properties of different solutions. We found some new exact solutions described with graphical representation showing solution wave structure, contour plot and wave propagation of the solution profile. The results are often helpful for studying the interaction of waves in many new localized structures and high-dimensional models.

Deep-learning of parametric partial differential equations from sparse and noisy data

Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete library, and spatially or temporally varying coefficients. In this work, a new framework, which combines neural network, genetic algorithm, and stepwise methods, is put forward to address all of these challenges simultaneously. In the framework, a trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solves the problem of sparse noisy data. Next, the genetic algorithm is used to discover the form of PDEs and corresponding coefficients, which solves the problem of the incomplete initial library. Finally, a stepwise adjustment method is introduced to discover parametric PDEs with spatially or temporally varying coefficients. In this method, the structure of a parametric PDE is first discovered, and then the general form of varying coefficients is identified. The proposed algorithm is tested on the Burgers equation, the convection-diffusion equation, the wave equation, and the KdV equation. Results demonstrate that this method is robust to sparse and noisy data, and is able to discover parametric PDEs with an incomplete initial library.

DL-PDE: Deep-Learning Based Data-Driven Discovery of Partial Differential Equations from Discrete and Noisy Data

In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, and then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by groundwater flow equation, convection-diffusion equation, Burgers equation and Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.

Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks

Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black–Scholes equation, whereas for the American option we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements). In the short note, previously published, a brief introduction to this work was given, where some ideas to price vanilla options by ANNs were presented, and only European options were addressed. In the current work, the methodology is introduced in much more detail.

Stochastic PDEs via convex minimization

We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approach via convex optimization to the approximation of nonlinear stochastic partial differential equations.

Heat transfer analysis of water based SWCNTs through parallel fins enclosed by square cavity

Analysis of natural convection due to various positions of bottom heated fin is performed in a square cavity that is filled by water-based Single Wall Carbon Nanotubes (SWCNTs) via finite element method (FEM). Three vertical parallel fins and one top horizontal fin is also placed inside the cavity. The upper surface of the cavity is adiabatic, however the rest of three parts of the square cavity are cold. Convection is driven through lower horizontal and vertically central heated fins. For convection, horizontal and middle vertical fins are heated with uniform temperature Th and remaining vertical parallel fins are fixed as cold (Tc). Mathematical structure is constructed in the form of system of nonlinear partial differential equations (PDEs) with constraint at the surface. Expressions of nanofluid relations are incorporated into the model. Effective thermal conductivity model includes the radius of nanoparticle and fluid molecules at nanoscale. Dimensionless form of PDEs are tackled through Galerkin technique-based Finite Element Method. Results are obtained for temperature profile and stream function that includes emerging parameters such as: Rayleigh number (Ra), nanoparticle fraction (ϕ), position of heated horizontal fin, variation in length forbottom heated fin (HT), Hartmann number (Ha) and effect of middle vertical fin (adiabatic, cold, hot). The analysis describes the significant effect of heat transfer in the presence of nanoparticle, heated length of bottom fin. Heat transfer rate increases by enlarging the heated length of the lower horizontal fin, and decreases by improving the value of Ra and solid volume fraction of nanoparticles.

Boundary Control Design for Traffic With Nonlinear Dynamics

We address a problem of boundary control for a nonlinear scalar conservation law. Namely, this article is devoted to the boundary control of a Lighthill-Whitham-Richards (LWR) partial differential equation (PDE) with triangular flux function evolving along a single road. The target state is a time- and space-dependent trajectory. The boundary control law is constructed using the analytical solution of the Hamilton–Jacobi (H–J) equation, which is an integral form of the LWR PDE. We design a feedback controller and illustrate its performance on a numerical example using the Godunov scheme.

Non-conservative variational approximation for nonlinear Schrödinger equations

In the work of Galley (Phys Rev Lett 110:174301, 2013) an initial value problem formulation of Hamilton’s principle was proposed and applied to non-conservative systems. Here, we explore this formulation for complex partial differential equations of the nonlinear Schrodinger (NLS) type, examining the dynamics of the coherent solitary wave structures of such models by means of a non-conservative variational approximation (NCVA). We compare the formalism of the NCVA to two other variational techniques used in dissipative systems, namely the perturbed variational approximation and a generalization of the so-called Kantorovich method. All three variational techniques produce equivalent equations of motion for the perturbed NLS models studied herein. We showcase the relevance of the NCVA method by exploring test case examples within the NLS setting including combinations of linear and density-dependent loss and gain. We also present an example applied to exciton–polariton condensates that intrinsically feature loss and a spatially dependent gain term.

Development of a compositional reservoir simulator for chemical enhanced oil recovery processes

To make decisions and ensure successful field development, oil companies use numerical simulation tools, including hydrodynamic simulators. This work is devoted to the development of a reservoir simulator based on a new formulation of partial differential equations of the chemical compositional model, as well as its testing by comparing the results of traditional and ASP flooding simulation with similar results of well-known simulators.

Bifractional Black-Scholes Model for Pricing European Options and Compound Options

AbstractRecent empirical studies show that an underlying asset price process may have the property of long memory. In this paper, it is introduced the bifractional Brownian motion to capture the underlying asset of European options. Moreover, a bifractional Black-Scholes partial differential equation formulation for valuing European options based on Delta hedging strategy is proposed. Using the final condition and the method of variable substitution, the pricing formulas for the European options are derived. Furthermore, applying to risk-neutral principle, we obtain the pricing formulas for the compound options. Finally, the numerical experiments show that the parameter HK has a significant impact on the option value.

About Some Possible Implementations of the Fractional Calculus

We present a partial panoramic view of possible contexts and applications of the fractional calculus. In this context, we show some different applications of fractional calculus to different models in ordinary differential equation (ODE) and partial differential equation (PDE) formulations ranging from the basic equations of mechanics to diffusion and Dirac equations.

An energy-based coupling approach to nonlocal interface problems

Nonlocal models provide accurate representations of physical phenomena ranging from fracture mechanics to complex subsurface flows, settings in which traditional partial differential equation models fail to capture effects caused by long-range forces at the microscale and mesoscale. However, the application of nonlocal models to problems involving interfaces, such as multimaterial simulations and fluid-structure interaction, is hampered by the lack of a physically consistent interface theory which is needed to support numerical developments and, among other features, reduces to classical models in the limit as the extent of nonlocal interactions vanish. In this paper, we use an energy-based approach to develop a formulation of a nonlocal interface problem which provides a physically consistent extension of the classical perfect interface formulation for partial differential equations. Numerical examples in one and two dimensions validate the proposed framework and demonstrate the scope of our theory.

Stability of Galerkin discretizations of a mixed space–time variational formulation of parabolic evolution equations

Abstract We analyze Galerkin discretizations of a new well-posed mixed space–time variational formulation of parabolic partial differential equations. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space–time discretization methods introduced by Andreev (2013, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33, 242–260) and by Steinbach (2015, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math., 15, 551–566).