Partial Differential Equations In Physics Research Articles

Numerical simulation and investigation of soliton solutions and chaotic behavior to a stochastic nonlinear Schrödinger model with a random potential.

The stochastic nonlinear Schrödinger model (SNLSM) in (1+1)-dimension with random potential is examined in this paper. The analysis of the evolution of nonlinear dispersive waves in a totally disordered medium depends heavily on the model under investigation. This study has three main objectives. Firstly, for the SNLSM, derive stochastic precise solutions by using the modified Sardar sub-equation technique. This technique is efficient and intuitive for solving such models, as shown by the generated solutions, which can be described as trigonometric, hyperbolic, bright, single and dark. Secondly, for obtaining numerical solutions to the SNLSM, the algorithms described here offer an accurate and efficient technique. Lastly, investigate the phase plane analysis of the perturbed and unperturbed dynamical system and the time series analysis of the governing model. The results show that the numerical and analytical techniques can be extended to solve other nonlinear partial differential equations in physics and engineering. The results of this study have a significant impact on how well we comprehend how solitons behave in physical systems. Additionally, they may serve as a foundation for the development of improved numerical techniques for handling challenging nonlinear partial differential equations.

Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics

Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.

A Comparative Analysis of Fractional-Order Fokker–Planck Equation

The importance of partial differential equations in physics, mathematics and engineering cannot be emphasized enough. Partial differential equations are used to represent physical processes, which are then solved analytically or numerically to examine the dynamical behaviour of the system. The new iterative approach and the Homotopy perturbation method are used in this article to solve the fractional order Fokker–Planck equation numerically. The Caputo sense is used to characterize the fractional derivatives. The suggested approach’s accuracy and applicability are demonstrated using illustrations. The proposed method’s accuracy is expressed in terms of absolute error. The proposed methods are found to be in good agreement with the exact solution of the problems using graphs and tables. The results acquired using the given approaches are also obtained at various fractional orders of the derivative. It is observed from the graphs and tables that fractional order solutions converge to an integer solution when the fractional orders approach the integer-order of the problems. The tabular and graphical view for the given problems is obtained through Maple. The presented approaches can be applied to existing non-linear fractional partial differential equations due to their accurate, simple and straightforward implementation.

RETRACTED: A novel approach for EMHD Williamson nanofluid over nonlinear sheet with double stratification and Ohmic dissipation

The current article highlights the non-Newtonian Williamson nanofluid with electro-magnetohydrodynamic (EMHD) flow over a nonlinear expanding sheet. Thermal and solutal stratification effects are considered due to the higher temperature difference and the impact of variable viscosity along with Ohmic dissipation is also incorporated. Transformation is applied for the conversion of physical partial differential equations (PDEs) into non-dimensional higher order nonlinear ordinary differential equations (ODEs). A well-known analytical approach known as the homotopy analysis method (HAM) is effectively applied to solve the differential equations. Different non-dimensional emerging parameters such as Weissenberg and Hartman number, Brownian motion and stratification parameters, stretching index, viscosity parameter, and Lewis number are used to check their impacts on velocity, concentration, and temperature profiles. To acquire the optimal solution through HAM, [Formula: see text] -curves are drawn. In the tabulated form, the numerical values for the non-dimensional Nusselt number and skin friction are arranged.

Predicting quantum potentials by deep neural network and metropolis sampling

The hybridizations of machine learning and quantum physics have caused essential impacts to the methodology in both fields. Inspired by quantum potential neural network, we here propose to solve the potential in the Schrödinger equation provided the eigenstate, by combining Metropolis sampling with deep neural network, which we dub as Metropolis potential neural network (MPNN). A loss function is proposed to explicitly involve the energy in the optimization for its accurate evaluation. Benchmarking on the harmonic oscillator and hydrogen atom, MPNN shows excellent accuracy and stability on predicting not just the potential to satisfy the Schrödinger equation, but also the eigen-energy. Our proposal could be potentially applied to the ab-initio simulations, and to inversely solving other partial differential equations in physics and beyond.

Penalization for a PDE with a nonlinear Neumann boundary condition and measurable coefficients

We consider a system of semilinear partial differential equations (PDEs) with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized PDEs, which converges to our initial problem. Since the coefficients we consider may be discontinuous, we use the notion of solution in the [Formula: see text]-viscosity sense. The method we use is based on backward stochastic differential equations and their S-tightness. This work is motivated by the fact that many PDEs in physics have discontinuous coefficients. As a consequence, it follows that if the uniqueness holds, then the solution can be constructed by a penalization.

A new approach to mathematical models of Drinfeld-Sokolov-Wilson and coupled viscous Burgers’ equations in water flow

In this paper, it is the first time that we implement conformable Laplace decomposition method (CLDM) to time fractional systems of Drinfeld-Sokolov-Wilson equation (DSWE) and coupled viscous Burgers’ equation (CVBE). DSWE and CVBE have an important place for cceanic, coastal sea research and they are considered as a mathematical model for shallow water waves and hydrodynmic turbulence respectively. At the end, the obtained solutions are compared with the exact solutions by the aid of tables and figures. The obtained results show that,conformable Laplace decomposition method (CLDM) is efficient, reliable, easy to apply and it gives researchers a new perspective for solving a wide variety of nonlinear fractional partial differential equations in physics.

On the new exact traveling wave solutions of the time-space fractional strain wave equation in microstructured solids via the variational method

In this paper, we mainly study the time-space fractional strain wave equation in microstructured solids. He’s variational method, combined with the two-scale transform are implemented to seek the solitary and periodic wave solutions of the time-space strain wave equation. The main advantage of the variational method is that it can reduce the order of the differential equation, thus simplifying the equation, making the solving process more intuitive and avoiding the tedious solving process. Finally, the numerical results are shown in the form of 3D and 2D graphs to prove the applicability and effectiveness of the method. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations in physics.

First Integral Method for Constructing New Exact Solutions of The important Nonlinear Evolution Equations in Physics

In this paper, some new exact solutions of the important nonlinear partial differential equations in physics as Gardener’s equation and Sharma-Tasso-Over equation are formally derived by utilizing the first integral method, where it is equipment us with many exact solutions by using the travelling wave transform, then deduce a system of ordinary differential equations which is solved by depending on theorem in commutative algebra and with helping the mathematical software like Maple and Wolfram Mathematica.

Finite element modelling of geophysical electromagnetic data with goal-oriented hr-adaptivity

Large discontinuities (jumps) in coefficients often appear when modelling physical problems involving inhomogeneous media. An example of this is the geophysical electromagnetic (EM) problem, where these jumps occur at interfaces which separate regions of (high) conductivity contrasts. These interfaces, along with other problem features, such as singular EM sources, motivate the use of adaptive mesh refinement to improve the efficiency of the solution of forward problems. Also, pointwise observations are made in geophysical EM surveys, which motivates the use of goal-oriented mesh adaptivity. In this work, we study the combined application of h- and r-refinements for the modelling of geophysical EM data. The proposed hr-adaptivity algorithm is thoroughly investigated in 1D, and then extended, and numerically validated, in 2D, using simple examples as well as a realistic model with irregular interfaces. In 1D, the steady-state diffusion and Helmholtz equations, which are commonly solved for the EM scalar and vector potentials, respectively [16], constitute the physical partial differential equations (PPDEs). In 2D, the Helmholtz equation is used as the PPDE. Additionally, a real 2D problem with a benchmark model is considered where the transverse electric (TE) mode of Maxwell’s equations is used as the PPDE. The r-refinements are based on an equidistribution principle and variational methods, for the 1D and 2D cases, respectively, which lead to mesh PDEs (MPDEs). The coupled PPDE and MPDE are solved in an iterative manner to enhance the accuracy of the PPDE solution by improving the equidistribution of a monitor function based on a posteriori error estimates. The results, in 1D, with two hierarchical error–based monitor functions and a residual error–based monitor function display the similarity between these functions in terms of the accuracy that could be achieved. In both 1D and 2D, comparisons are made between global and goal-oriented adaptivity which show the advantage of goal-oriented error estimates in gaining higher accuracy at a target, compared to global error estimates. In 2D, the results also demonstrate the higher efficiency of hr-adaptivity compared with pure h-refinement, in terms of computation time.

High order discontinuous Galerkin methods for time-domain and frequency-domain nanophotonics

The discontinuous Galerkin (DG) method is a general numerical modeling approach that has been extensively studied in the last 20 years for the solution of many systems of partial differential equations in physics. Its development for the numerical treatment of the system of Maxwell equations was initiated by the applied mathematics community in the early 2000s. It is now a very popular method for time-domain electromagnetics, which is increasingly used and further developed by the applied physics and electrical engineering communities. More recently, a specific variant of DG, referred to as hybridized DG (or HDG), has been proposed for frequency-domain electromagnetics. DG methods possess nice features that make them particularly attractive for dealing with heterogeneous media and irregularly shaped or curved geometries, and more generally with multiscale problems. Not surprisingly, the method has also been adopted by researchers in the nano-optics field. In this paper, we report on our recent efforts for extending the capabilities of this family of methods for the numerical modeling of nanoscale light-matter interactions.

New travelling solitary wave solutions for an evolution equation by three schemes

In this study, we establish new travelling wave Solitons solutions by using three methods. These methods are Tan-Cot method, Extended Tanh-Coth method, and Modified Simple equation Method. These methods are used to obtain new solitary wave trigonometric and hyperbolic functions solutions for the generalized Schamel-Korteweg-de Vries (S-KdV) equation. These methods have been successfully applied to construct new solitary solutions as illustrated in figures. The three methods are efficient and reliable for solving great many nonlinear partial differential equations in physics.

Cell Detection by Functional Inverse Diffusion and Non-negative Group Sparsity—Part I: Modeling and Inverse Problems

In this two-part paper, we present a novel framework and methodology to analyze data from certain image-based biochemical assays, e.g., ELISPOT and Fluorospot assays. In this first part, we start by presenting a physical partial differential equations (PDE) model up to image acquisition for these biochemical assays. Then, we use the PDEs' Green function to derive a novel parametrization of the acquired images. This parametrization allows us to propose a functional optimization problem to address inverse diffusion. In particular, we propose a non-negative group-sparsity regularized optimization problem with the goal of localizing and characterizing the biological cells involved in the said assays. We continue by proposing a suitable discretization scheme that enables both the generation of synthetic data and implementable algorithms to address inverse diffusion. We end Part I by providing a preliminary comparison between the results of our methodology and an expert human labeler on real data. Part II is devoted to providing an accelerated proximal gradient algorithm to solve the proposed problem and to the empirical validation of our methodology.

Steady-state temperature distribution in a tetragonal solid

ABSTRACTThe main objective of the project is to use a simple numerical technique to find out the temperature distribution in each of the three planes of the block. For many practical problems, numerical methods to solve partial differential equations (PDEs) are required. Conventional finite element or finite-difference codes have a difficulty to obtain precise solutions because of the need for an exceedingly fine mesh, which leads to often prohibitive CPU time. While conventional methods exhibit such a difficulty, some practical problems even require solutions guaranteed. The Laplace equation is one of the important PDEs in physics and engineering, describing the phenomenology of electrostatics among others, and various problems for the Laplace equation require highly precise and verified solutions. We present an alternative approach based on numerical method by computer programming in order to find the temperature distribution in the three-dimensional solid.

Mesh Spacing Estimates and Efficiency Considerations for Moving Mesh Systems

AbstractAdaptive numerical methods for solving partial differential equations (PDEs) that control the movement of grid points are called moving mesh methods. In this paper, these methods are examined in the case where a separate PDE, that depends on a monitor function, controls the behavior of the mesh. This results in a system of PDEs: one controlling the mesh and another solving the physical problem that is of interest. For a class of monitor functions resembling the arc length monitor, a trade off between computational efficiency in solving the moving mesh system and the accuracy level of the solution to the physical PDE is demonstrated. This accuracy is measured in the density of mesh points in the desired portion of the domain where the function has steep gradient. The balance of computational efficiency versus accuracy is illustrated numerically with both the arc length monitor and a monitor that minimizes certain interpolation errors. Physical solutions with steep gradients in small portions of their domain are considered for both the analysis and the computations.

The Maxwell equations as a Bäcklund transformation

Bäcklund transformations (BTs) are a useful tool for integrating nonlinear partial differential equations (PDEs). However, the significance of BTs in linear problems should not be ignored. In fact, an important linear system of PDEs in Physics, namely, the Maxwell equations of Electromagnetism, may be viewed as a BT relating the wave equations for the electric and the magnetic field, these equations representing integrability conditions for solution of the Maxwell system. We examine the BT property of this system in detail, both for the vacuum case and for the case of a linear conducting medium.

Schrödinger’s radial equation

The August 2014 issue of Physics Today contains a letter (page 8) in which M. Y. Amusia comments on the article “Bohr’s molecular model, a century later” by Anatoly Svidzinsky, Marlan Scully, and Dudley Herschbach (Physics Today, January 2014, page 33). Amusia points out that the radial part of the article’s D-dimensional Schrödinger equation in Hartree units isand he criticizes the implication that the Coulomb potential does not depend on D, which it surely does. Half a century ago, following up on the work of Paul Ehrenfest,11. P. Ehrenfest, Proc. Amsterdam Acad. 20, 200 (1917); Ann. Phys. Leipzig 61, 440 (1920). who had studied the hydrogen atom in n spatial dimensions using Bohr orbit theory, I reformulated the problem22. F. R. Tangherlini, Nuovo Cimento 27, 636 (1963). https://doi.org/10.1007/BF02784569 using Schrödinger’s equation extended to n dimensions, in which I had the Coulomb potential for n ≥ 3. I did not include the Coulomb potential for n = 1,2, as discussed by Amusia, as I was only interested in the stability of the higher-dimensional atom for n > 3. (For the dimensionality of space I used “n” rather than “D.”) In that work I gave the radial equation equivalent to the one above: I wrote, “If we now transform to n-dimensional polar coordinates, introduce n-dimensional spherical harmonics, and factor out the angular dependence,33. See, for example, A. Sommerfeld, Partial Differential Equations in Physics, E. G. Straus, trans., Academic Press (1949), app. 4. the resulting radial equation takes the formIn my paper, I did not eliminate the dR/dr term to arrive at a form equivalent to the first equation. However, that is readily done by setting R = r−(n − 1)/2ϕ and generalizing to nuclear charge Z, and setting n → D. The reason I have the D-dependent Coulomb potential for D > 3 and the article authors do not is that they have a different goal—to gain new insight into the real, atomic-molecular 3D world using the limiting behavior of the D-dimensional kinetic energy. In contrast, as indicated above, I was only interested in whether the D-dimensional Schrödinger “hydrogen atom” would have stable bound states for D > 3.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. P. Ehrenfest, Proc. Amsterdam Acad. 20, 200 (1917); Google ScholarAnn. Phys. Leipzig 61, 440 (1920). Google Scholar2. F. R. Tangherlini, Nuovo Cimento 27, 636 (1963). https://doi.org/10.1007/BF02784569, Google ScholarCrossref3. See, for example, A. Sommerfeld, Partial Differential Equations in Physics, E. G. Straus, trans., Academic Press (1949), app. 4. Google Scholar© 2015 American Institute of Physics.

On some nonlinear fractional PDEs in physics

In this paper, we applied relatively new fractional complex transform (FCT) to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and Variational Iteration Method (VIM) is to find approximate solution of time- fractional Fornberg-Whitham and time-fractional Wu-Zhang equations. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs arising in mathematical physics and hence can be extended to other problems of diversified nonlinear nature. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the proposed algorithm. DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11687BIBECHANA 12 (2015) 59-69

New Traveling Wave Solutions of the Boussinesq Equation Using a New Generalized Mapping Method

In this paper, a generalized mapping method for finding the exact traveling wave solutions of a nonlinear partial differential equation is discussed. Firstly, some new solutions of an auxiliary ordinary differential equation are introduced. They are then used to generate new exact solutions for the Boussinesq equation. The new solutions are then grouped into ten families and the properties of each family of solutions are demonstrated. We should also emphasize here that the developed method can also be applied to a large variety of nonlinear partial differential equations in physics and mechanics.

Travelling wave solutions of the generalized Zakharov-Kuznetsov equation via the extended generalized Riccati equation mapping method

Exact hyperbolic, trigonometric and rational travelling wave solutions to the (2+1)-dimensional Generalized Zakharov-Kuznetsov (GZK) equation via the extended generalized Riccati equation mapping method are presented in this paper. The twenty one travelling wave solutions obtained were verified by putting them back into the GZK equation with the aid of Mathematica. This shows that the extended generalized Riccati equation mapping method is a powerful tool for finding exact solution to nonlinear partial differential equations in physics, mathematics and other applications.