Solution Of Partial Differential Equations Research Articles

Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

The use of R code for modeling and simulation of 2-D contaminant migration through a soil for the pulse source

<p>In this study, based on an existing analytical solution for the two-dimensional transport of contaminants in a saturated soil layer, for the pulse source, the R program code was developed. The simulation is used to obtain the profiles of contaminant concentration, for a steady groundwater velocity, as a function of distance from the source and time. The problem is analytically solved by leveraging the similarity between the Gaussian (normal) distribution and contaminant concentration distribution, the development of the analytical model (i.e., solution of partial differential equation) by using this similarity is explained step by step since it may be of interest to researchers. Contaminant propagation is modeled using R software, which helps to understand how contaminants migrate through a saturated soil layer. This approach aids in comprehending the mechanisms and spatial dynamics of contaminant dispersion, facilitating the prediction and management of soil and groundwater contamination. The provided R code can be altered for different parameters and time intervals.</p>

The use of R code for modeling and simulation of 2-D contaminant migration through a soil for the pulse source

<p>In this study, based on an existing analytical solution for the two-dimensional transport of contaminants in a saturated soil layer, for the pulse source, the R program code was developed. The simulation is used to obtain the profiles of contaminant concentration, for a steady groundwater velocity, as a function of distance from the source and time. The problem is analytically solved by leveraging the similarity between the Gaussian (normal) distribution and contaminant concentration distribution, the development of the analytical model (i.e., solution of partial differential equation) by using this similarity is explained step by step since it may be of interest to researchers. Contaminant propagation is modeled using R software, which helps to understand how contaminants migrate through a saturated soil layer. This approach aids in comprehending the mechanisms and spatial dynamics of contaminant dispersion, facilitating the prediction and management of soil and groundwater contamination. The provided R code can be altered for different parameters and time intervals.</p>

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

Gyrator potential operator and Lp -Sobolev spaces involving Gyrator transform

This paper presents the properties of Gyrator potential operator G μ l and related L p -Sobolev spaces associated with Gyrator transform (GT). The Schwartz-type space S Δ ( R 2 ) is introduced. Various properties of the kernel of GT is studied including its continuity. Pseudo-differential operators A α , q associated with GT is defined and derived it's continuity on S Δ ( R 2 ) . The Gyrator potential operator G μ l is defined as a pseudo-differential operator associated with a precise symbol and obtained certain fruitful properties. The operator G μ l is extended to a space of distributions. Another integral representation of A α , q is obtained and its L p ( R 2 ) -boundedness result is derived. The spaces H α m , p ( R 2 ) and H Δ m , p ( R 2 ) are defined. It is shown that the operator G μ l is an isometry of H Δ m , p ( R 2 ) . An L p -boundedness result for the operator G μ l is proved. We conclude the article by applying some of the results to show that the analytical solution of certain non-homogeneous partial differential equations belong to these spaces.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

A novel least squares approach generating approximations orthogonal to the null space of the operator

We introduce a novel least squares functional specifically formulated to solve linear partial differential equations with operators that have a nonempty null space. Our method involves projecting the solution onto the orthogonal complement of the operator's null space to overcome challenges encountered by conventional numerical methods when nonzero null components are present. We describe the theoretical framework of the proposed method and validate it through numerical examples that show improved accuracy and usability in cases where traditional methods are less effective due to significant null space components. Overall, this approach provides a practical and reliable solution for partial differential equations with substantial null space components.

Combining approach of collocation and finite difference methods for fractional parabolic PDEs

This research aims to estimate the solutions of fractional-order partial differential equations of spacial fractional and both time-space fractional order. For this, we use finite differences for time derivatives and the well-known collocation method for space derivatives with lower-order Bernstein polynomials as basis functions. We explain the mathematical formulations in detail. Convergence and stability analysis of the space–time fractional diffusion equation with the source term is reported subsequently. Three numerical examples are considered for demonstrating the accuracy and reliability of the proposed method.

Weak, Strong and Global Solutions for DNA Model with Memory

Objective: To determine solutions of hyperbolic partial differential equations with memory using operator techniques with the corresponding estimates. The hyperbolic model associated with the Peyrard-Bishop model of DNA with memory is especially analyzed. Method: Using Gronwall inequalities and operator theory, solutions to partial differential equations are obtained after reduction to an equivalent one. Results and Discussion: The existence of three types of weak, strong and global solutions for the hyperbolic model associated with the Peyrard-Bishop model of DNA with memory has been demonstrated. Implications of the research: Apply in the first university years the application of MATLAB and Python to partial differential equations in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This paper also provides an alternative approach to Analysis of DNA vibration amplitudes with past-dependent memory.

Results on solutions of several product type nonlinear partial differential equations in [formula omitted]

This paper is devoted to exploring the solutions of the product type nonlinear partial differential equations (PDEs) with three complex variables. By making use of Nevanlinna theory with several complex variables and Hadamard factorization theory of meromorphic functions, and combining with the properties of the full rank determinants and algebraic cofactor, we prove that equation∏i=13(ai1uz1+ai2uz2+ai3uz3)=1 in C3 has no any finite order transcendental entire solution under the condition that the rank of the following matrixA=(a11a12a13a21a22a23a31a32a33) is full rank, i.e. R(A)=3, where aij(i,j=1,2,3) are constants in C. Our results are some improvements and generalizations of the previous results given by Saleeby, Li, Lü and Xu. Meantime, we list some examples to explain that the condition in our theorem can not be removed.

Early-time resonances in the three-dimensional wall-bounded axisymmetric Euler and related equations

We investigate the complex-time analytic structure of solutions of the three-dimensional (3D)-axisymmetric, wall-bounded, incompressible Euler equations, by starting with the initial data proposed in Luo and Hou [“Potentially singular solutions of the 3D axisymmetric Euler equations,” Proc. Natl. Acad. Sci. U. S. A. 111(36), 12968–12973 (2014)], to study a possible finite-time singularity. We use our pseudospectral Fourier–Chebyshev method [Kolluru et al., “Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,” Phys. Rev. E 105(6), 065107 (2022)], with quadruple-precision arithmetic, to compute the time-Taylor-series coefficients of the flow fields, up to a high order. We show that the resulting approximations display early-time resonances; the initial spatial location of these structures is different from that for the tygers, which we have obtained in Kolluru et al. [“Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,” Phys. Rev. E 105(6), 065107 (2022)]. We then perform asymptotic analysis of the Taylor-series coefficients, by using generalized ratio methods, to extract the location and nature of the convergence-limiting singularities and demonstrate that these singularities are distributed around the origin, in the complex-t2 plane, along two curves that resemble the shape of an eye. We obtain similar results for the one-dimensional (1D) wall-approximation (of the full 3D-axisymmetric Euler equation) called the 1D Hou–Luo model, for which we use Fourier-pseudospectral methods to compute the time-Taylor-series coefficients of the flow fields. Our work examines the link between tygers, in Galerkin-truncated pseudospectral studies, and early-time resonances, in truncated time-Taylor expansions of solutions of partial differential equations, such as those we consider.

Shape and topology optimization method with generalized topological derivatives

This paper introduces a novel method for shape and topology optimization based on a generalized approach for evaluating topological derivatives, which are essential for the integration of shape and topology optimization. Traditionally, evaluating these derivatives presents significant mathematical challenges due to the discontinuity introduced by the insertion of a hole within the domain of interest. To overcome this issue, the study employs Helmholtz-type partial differential equations (PDEs) to construct a filtered objective functional. This approach ensures differentiability across the material and void phases and continuity over the fixed design domain while maintaining the same evaluation value as the original objective functional. By considering differentiability, continuity conditions, and the relationship between shape and topological derivatives during asymptotic analysis, generalized topological derivatives are obtained through established mathematical procedures. These topological derivatives exhibit a direct correlation with the PDE solutions and demonstrate satisfactory smoothness, thereby facilitating refined shapes in optimization strategies. Furthermore, an effective shape update algorithm is proposed, which directly integrates topological derivatives into structural optimization problems, simplifying their implementation and improving efficiency. Finally, the efficacy of the proposed methodology is demonstrated through its application to various optimal design problems, including stiffness maximization, compliant mechanisms, and eigenfrequency maximization. Verification results further highlight its potential to enhance existing methods for addressing more practical and complex optimization challenges.

Mathematical Models of Diffusion in Physiology.

Diffusion is a mass transport phenomenon caused by chaotic thermal movements of molecules. Studying the transport in specific domain is simplified by using evolutionary differential equations for local concentration of the molecules instead of complete information on molecular paths [1]. Compounds in a fluid mixture tend to smooth out its spatial concentration inhomogeneities by diffusion. Rate of the transport is proportional to the concentration gradient and coefficient of diffusion of the compound in ordinary diffusion. The evolving concentration profile c(x,t) is then solution of evolutionary partial differential equation deltac/deltat=DDeltac where D is diffusion coefficient and Delta is Laplacian operator. Domain of the equation may be a region in space, plane or line, a manifold, such as surface embedded in space, or a graph. The Laplacian operates on smooth functions defined on given domain. We can use models of diffusion for such diverse tasks as: a) design of method for precise measurement of receptors mobility in plasmatic membrane by confocal microscopy [2], b) evaluation of complex geometry of trabeculae in developing heart [3] to show that the conduction pathway within the embryonic ventricle is determined by geometry of the trabeculae.

A new study on fractional Schamel Korteweg–De Vries equation and modified Liouville equation

Traveling wave solutions of fractional partial differential equations have great importance in the literature. The diversity of solutions plays an important role in understanding the physical structure of the model it represents. For this reason, two important differential equations with the fractional order, which have a significant role in applied sciences and can model real-life problems most accurately, have been solved by the generalized (G′G)-expansion method in this study. This method is a generalization of the classical (G′G)-expansion method. With this developed method, the non-linear fractional Schmael Korteweg–De Vries equation and fractional modified Liouville differential equations are discussed to find their exact solutions. In this way, new exact solutions of these equations that were not previously included in the literature have been found. The presented method has been applied to these two equations for the first time, and various new traveling wave solutions have been obtained. Thus, the study goes beyond other studies. To understand the physical behavior of these new exact solutions, three-dimensional graphs have been drawn according to different parameter values.

Exploring solutions to the fractional Boiti–Leon–Manna–Pempinelli equation characterizing wave dynamics in incompressible fluids

In this study, the (3 + 1)-dimensional space-time fractional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated utilizing the Kudryashov method (KM) and the modified Kudryashov method (MKM). These two efficient methods are implemented to acquire exact closed-form solutions to the considered fractional BLMP equation, and novel solitary wave solutions are constructed involving hyperbolic functions, and exponential rational functions with arbitrary constant parameters. With the aid of a unique wave transformation, the nonlinear fractional differential equation is transformed into an ordinary differential equation. In this unique wave transformation, the conformable fractional derivative is considered to which the chain rule is applied. The obtained solutions are indeed beneficial for analyzing the dynamic behavior of the fractional BLMP equation in describing fluid propagation. A few three-dimensional representations of the attained exact analytic solutions are offered by accounting for the proper selection of the appropriate parameters with the help of mathematical software. As a result, novel soliton solutions, such as the anti-kink type and singular kink type waves, are acquired from the attained solutions. This approach is relatively straightforward and efficient, making it possible to utilize it to obtain exact solutions for higher-order nonlinear partial differential equations.

Stable weight updating: A key to reliable PDE solutions using deep learning

Deep learning techniques, particularly neural networks, have revolutionized computational physics, offering powerful tools for solving complex partial differential equations (PDEs). However, ensuring stability and efficiency remains a challenge, especially in scenarios involving nonlinear and time-dependent equations. This paper introduces novel residual-based architectures, namely the Simple Highway Network and the Squared Residual Network, designed to enhance stability and accuracy in physics-informed neural networks (PINNs). These architectures augment traditional neural networks by incorporating residual connections, which facilitate smoother weight updates and improve backpropagation efficiency. Through extensive numerical experiments across various examples—including linear and nonlinear, time-dependent and independent PDEs—we demonstrate the efficacy of the proposed architectures. The Squared Residual Network, in particular, exhibits robust performance, achieving enhanced stability and accuracy compared to conventional neural networks. These findings underscore the potential of residual-based architectures in advancing deep learning for PDEs and computational physics applications.

Numerical approximations of a lattice Boltzmann scheme with a family of partial differential equations

In this contribution, we address the numerical solutions of high-order asymptotic equivalent partial differential equations with the results of a lattice Boltzmann scheme for an inhomogeneous advection problem in one spatial dimension. We first derive a family of equivalent partial differential equations at various orders, and we compare the lattice Boltzmann experimental results with a spectral approximation of the differential equations. For an unsteady situation, we show that the initialization scheme at a sufficiently high order of the microscopic moments plays a crucial role to observe an asymptotic error consistent with the order of approximation. For a stationary long-time limit, we observe that the measured asymptotic error converges with a reduced order of precision compared to the one suggested by asymptotic analysis.

Traffic assignment optimization to improve urban air quality with the unified finite-volume physics-informed neural network

Emissions by road transportation constitute the major contributor of air pollutants to threat the public health, especially on mega cities with high-rise buildings and growing population. A practicable and economical remedy is assigning the traffic volume judiciously, termed traffic assignment optimization (TAO). Incorporating the computational fluid dynamics (CFD), this method provides precisely optimized traffic volumes under versatile meteorological conditions. However, the high-dimensional degree of freedoms (DoF) involved in CFD hampers its further application toward the large-scale problem where a massive urban area is considered. With the more complicated urban traffic network, the larger decision variable dimension also impedes the time-efficient acquisition of the optimization results. To alleviate this curse-of-dimensionality, the current work proposes an optimization framework with the surrogate model based on the unified finite-volume physics-informed neural networks (UFV-PINN). The UFV-PINN plays the dual role as the partial differential equation (PDE) solver and the surrogate model for pollutant concentration prediction. It reduces the DoF within the PDE solution process and enables the gradient-based optimization. The current work optimizes the average CO concentration and accumulative travel time in Kowloon peninsula of Hong Kong, using the multiple-gradient descent algorithm (MGDA). This optimization framework is tested with various working conditions. Results indicate that the proposed method attain high accuracy solutions, comparable to the heuristic algorithms. This study is the first attempt to incorporate UFV-PINN into the TAO with air quality consideration. Endowed with the differentiability by UFV-PINN, the framework will facilitate the TAO to provide precise suggestions toward large-scale problems.

Stability criterion of a nonautonomous 3-species ratio-dependent diffusive predator-prey model

The global stability of a nonautonomous 3-species ratio-dependent diffusive predator-prey model is studied in this paper. Firstly, some easily verifiable sufficient conditions which guarantee the existence of the strictly positive space homogenous periodic solution (SHPS) of the ratio- dependent predator-prey model (RDPPM) with diffusive and variable coefficient are achieved by using a comparison theorem of differential equation and fixed point theorem. At the same time, some new analysis techniques are developed as a byproduct. Secondly, some sufficient conditions ensuring the globally asymptotically stability of the strictly positive SHPS of the diffusive nonautonomous predator-prey model are given by using the method of upper and lower solutions (UALS) for the parabolic partial differential equations and Lyapunov stability theory. In addition, two numerical examples are given to validate the theoretical results obtained in this paper.