Partial Differential Equations in Applied Mathematics最新文献

This note aims to study the existence of the local solutions and derive a blow-up result for a quasi-linear bi-hyperbolic equation with dynamic boundary conditions. We use the maximal monotone operator theory to demonstrate the solution’s local well-posedness, and a concavity method to establish the blow-up result.

We propose a Lie group method to discuss the modified KP equation appearing in Thomas–Fermi (TM) plasma, characterised by cold and hot electrons. The Lie method facilitates the identification of similarity reductions, infinitesimal symmetries, group-invariant solutions, and novel analytical solutions for nonlinear models. The similarity reduction method is carried out to transform the nonlinear partial differential equations (NLPDE) into the nonlinear ordinary differential equations (NLODE). This study focuses on solitary wave profiles due to their usefulness in various engineering applications, including monitoring public transportation systems, managing coastlines, and mitigating disaster risks. It also addresses the conservation laws associated with the modified KP equation. The generalised auxiliary equation (GAEM) scheme is used to compute the new solitary wave patterns of the modified KP equation, which explains the dynamics of nonlinear waves in Thomas–Fermi plasma. The idea of nonlinear self-adjointness is used to compute the conservation laws of the examined model. The graphical behaviour of some solutions is represented by adjusting the suitable value of the parameters involved.

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.

Introduced in Zhang et al. (2012), the trilinear Boussinesq equation is the natural form of the equation for the $\tau$-function of the lattice Boussinesq system. In this paper we study various aspects of this equation: its highly nontrivial derivation from the bilinear lattice AKP equation under dimensional reduction, a quadrilinear dual lattice equation, conservation laws, and periodic reductions leading to higher-dimensional integrable maps and their Laurent property. Furthermore, we consider a higher Gel’fand–Dikii lattice system, its periodic reductions and Laurent property. As a special application, from both a trilinear Boussinesq recurrence as well as a higher Gel’fand–Dikii system of three bilinear recurrences, we establish Somos-like integer sequences.

In this study, we investigate the analytical soliton solutions of a fundamental model, namely the nonlinear Schrödinger-Hirota equation, in the context of beta time-fractional derivative. We adopt the (ω′/ω,  1/ω)-expansion method, which is a reliable and straightforward approach to extract fresh and general soliton solutions in terms of hyperbolic, trigonometric, and rational functions. The solitons include anti-kink, anti-bell-shaped, bell-shaped, and periodic solitons. These solitons have significant applications in various scientific fields, such as optical fiber communications, signal processing, plasma physics, and trans-oceanic data transfer. This study demonstrates the significance of fractional-order differentiation in revealing new solitons. We also provide a comprehensive comparison with existing literature in normal and anomalous dispersion regions, highlighting the uniqueness of the solutions. Moreover, the graphical representations are used to illustrate the properties and potential applications of these solitons. This research might contribute to the advancement of nonlinear optical research and technology.

In this paper, the error analysis of the Petrov–Galerkin finite volume element method (FVEM) is investigated for a nonlinear parabolic integro-differential equation that arises in the mathematical modeling of the penetration of a magnetic field into a substance, accounting for temperature-dependent changes in electrical conductivity. Starting from Maxwell’s equations, we derive a one-dimensional model problem, which forms the basis of our analysis. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. For simplicity, we consider only the lowest-order (linear and L-splines) finite volume elements. The novel contribution lies in the application of FVEM to this problem, leading to the establishment of an unconditionally stable numerical scheme and the derivation of optimal error estimates in the $L^{\infty }\left(L^2\left(\Omega \right)\right)$ and $L^2\left(H_0^1\left(\Omega \right)\right)$ norms for both semi-discrete and linearized backward Euler fully-discrete schemes, using a generalized projection method that carefully manages the nonlinear terms. Lastly, numerical experiments are provided to support the theoretical conclusions.

In this paper, we investigate the fractional model of the Liénard and Duffing equations with the Liouville-Caputo fractional derivative. These equations grow with the evolution of radio and vacuum tube technology, which describe oscillating circuits and generalize the spring–mass device equation. We compare two numerical approaches, namely Jacobi and Haar wavelet collocation methods. The given approaches are used to discretize and transform the equation into a system of algebraic equations, and the Broyden-Quasi Newton algorithm is applied to solve the resulting nonlinear system of equations. A complete error analysis and convergence rates for different grid sizes are derived for both methods, which are used to compare the accuracy and efficiency of the two approaches. While both approaches produce correct solutions, according to the numerical findings, the Jacobi collocation method is more efficient and accurate than the Haar wavelet collocation method.

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

The current communication deliberates the consequences of the Darcy–Forchheimer flow of Eyring–Powell nanofluid past a slippery surface containing activation energy and motile microorganisms. The flow is influenced by the consequences of Brownian motion, thermal radiation, the Cattaneo–Christov heat-mass flux theory, and thermophoresis. The framed flow models are transformed into ordinary derivative equations by adopting appropriate conversion variables. The transformed equations are numerically tackled by using the bvp4c scheme in MATLAB. The study is remarkable for its comprehensive analysis of the interplay of several flow factors, such as the Forchheimer number, Richardson number, bioconvection Rayleigh number, radiation, thermophoresis, Brownian motion, thermal and mass relaxation time parameters. The outcomes are visualized through tables and diagrams, which provide significant insights into the intricate physical mechanisms involved in this multifaceted subject. Evidently, the velocity profile declines when there is a rise in the buoyancy ratio parameter and the opposite trend is obtained for the Richardson number. The temperature grows when there is a larger magnitude of the thermophoresis parameter and it reduces for greater values of the time relaxation parameter. The activation energy and mass relaxation parameters enhance the concentration profile. The microbe density increases when enhancing the quantity of Peclet number and it declines for bioconvection Lewis number.

The elasticity theory problem for a radially inhomogeneous cylinder of small thickness, whose elastic moduli are arbitrary continuous functions depending on the radius of the cylinder, is considered. It is assumed that the side surface of the cylinder is stress-free, and the boundary conditions that keep the cylinder in equilibrium are given at its seats. Asymptotic solutions are constructed by the asymptotic integration method. It is shown that the asymptotic solution consists of the sum of the penetrating solution, simple boundary effect and boundary layer solutions. The character of the stress-strain state corresponding to the penetrating solution, simple boundary effect and boundary layer solutions is determined. Asymptotic formulas are obtained for displacements and stresses, which allow to calculate the stress-strain state of a cylinder.

The problem of torsion of a radially inhomogeneous cylinder is studied, with its lateral surface free from stress and the boundary conditions keeping it in equilibrium at its seats. By applying the asymptotic integration method, it is determined that the asymptotic solution for the torsion problem consists of the sum of the penetrating solution and boundary layer solutions.

Numerical analysis is performed and the effect of material inhomogeneity on the stress-strain state of the cylinder is evaluated.