Solution Of Equation Research Articles

S-asymptotically $$\omega $$-periodic solutions for time-space fractional nonlocal reaction-diffusion equation with superlinear growth nonlinear terms

S-asymptotically $$\omega $$-periodic solutions for time-space fractional nonlocal reaction-diffusion equation with superlinear growth nonlinear terms

Solutions of fractional differential models by using Sumudu transform method and its hybrid

This paper presents the Sumudu transform method and its hybrid for the construction of solutions of differential equations, both with integer-order and fractional derivatives. The paper discusses the construction of solutions of fractional differential equations with varying delay proportional to the independent variable by using two methods. The paper considers the mathematical model for the decay of Iodine 135 as an application of fractional differential equations in nuclear physics. The application indicates that fractional differential equations with variable delay proportional to the independent variable are a useful tool for the modeling of many anomalous phenomena in nature and in the theory of complex systems.

Two-Level method for blind image deblurring problems

Blind image deblurring (BID) is a procedure for reducing blur and noise in a deteriorated image. In this process, the estimation of the original image, as well as the blurring kernel of the degraded image, is done without or with only partial information about the imaging system and degradation. This is an inverse problem (ill-posed) that corresponds to the direct problem of deblurring. To overcome the ill-posedness of BID and attain useful solutions, the regularization models based on mean curvature (MC) are utilized. The discretization of MC-based models often leads to a large ill-conditioned nonlinear system of equations, which is computationally expensive. Moreover, the existence of MC functionals in the governing equations of the BID model complicates the calculation of the nonlinear system. To overcome these problems, in this paper, we propose the Two-Level blind image deblurring method (TLBID). First, on the coarse-grid, we solve a small nonlinear system (with a small number of pixels) for a mesh size of H, followed by solving a large linear system of equations on the finer grid (with a large number of pixels) of size h (h≤H). On the coarse mesh, we solve the BID problem utilizing the computationally expensive MC regularization functional. After this, we interpolate the results to the finer mesh. On the finer mesh, we solve the BID problem with less computationally expensive regularization functionals such as total variation (TV) or Tikhonov. This approach produces an approximate solution of the BID equations with high accuracy, which is cost-effective. The TLBID algorithm is implemented with MATLAB, and verification and validation are carried out using benchmark problems and medical digital images.

Wave propagation in the Kolmogorov–Petrovskii–Piskunov–Fischer equation with time delay and spatial deviation

For the logistic equation with diffusion two generalizations are considered: one with time delay and the other with deviation in the spatial variable. For these equations, the problem of density wave propagation is considered. These equations, known as Kolmogorov–Petrovsky–Piskunov–Fisher equations with delay or deviation (KPPF with delay or deviation), are investigated using both asymptotic and numerical methods.To investigate the qualitative behavior of solutions of such equations, the wave profile equation is considered and the conditions for the emergence of its oscillatory modes are identified.The local properties of solutions with periodic boundary conditions are examines, revealing that an increase in the period leads to the emergence of stable solutions with a more complex spatial structures.Numerical analysis of the wave propagation process indicates that at sufficiently small delay values, this equation has solutions that are close to those of the standard KPPF equation. Increasing the delay parameter leads first to the appearance of a damped oscillatory component in the spatial distribution of the solution. With further increases of this parameter, a complex spatial profile of the wave appears.In the final part of the chapter, which focuses on numerical calculation, the numerical analysis of the wave propagation process from two initial perturbations is performed. This analysis tracks the process of wave interaction. The complex spatially inhomogeneous structure that arises during wave propagation and interaction can be explained by the properties of the corresponding solutions of the periodic boundary value problem, especially as the interval of change in the spatial variable increases.

Lattice Boltzmann simulation of plunging breakers

In the present work, the conservative phase-field lattice Boltzmann model is applied to simulate plunging breakers in shallow waters. The solver is computationally efficient as it is explicit in time stepping and does not require a solution for the pressure Poisson equation. Nevertheless, the solver can simultaneously handle a large Reynolds number ( = 10 4 ) and a high-density ratio of 1000. Simulations are performed by varying initial steepness and dispersion parameters, which influence the nonlinearity degree of wave (or the growth speed of instability) and the oscillatory motion (limited by the bottom) on the breaking characteristics. Detailed analyses of kinematics, air entrainment, vorticity, and energy dissipation are presented.

On the critical points of solutions of PDE in non-convex settings: The case of concentrating solutions

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely{−Δu=λeu in Ωu=0 on ∂Ω, where Ω⊂R2 is a bounded smooth domain and λ>0 is a small parameter.

Power series solution for fractal differential equations

Power series solution for fractal differential equations

Bifurcation analysis and solitary wave solution of fractional longitudinal wave equation in magneto-electro-elastic (MEE) circular rod

In this study, we investigate the longitudinal wave equation (LWE) with the M−fractional derivative, which describes the propagation of longitudinal waves along a rod while incorporating interactions between mechanical, electrical, and magnetic fields within the material. Initially, we apply bifurcation analysis to examine the critical points or phase portraits where the system transitions to new behaviors, such as stability shifts or the emergence of chaos, and observe the mechanism of static soliton formation through a saddle-node bifurcation. Subsequently, we utilize a modified simple equation (MSE) technique to find solitary wave solutions. Depending on the relationships between free parameters, the solutions are expressed as hyperbolic, trigonometric, and exponential functions. The numerical form of the obtained solutions reveals complex phenomena, including dark and bright bell waves, kink periodic lump waves, kink periodic waves, periodic lump waves, interaction waves between kink and periodic waves, linked lump waves, and interactions of periodic and lump waves. Additionally, we compare our results with previously published work, demonstrating that the discussed methods are valuable tools for providing distinct, accurate soliton solutions relevant to nonlinear science and technology applications.

A multilevel Monte Carlo algorithm for stochastic differential equations driven by countably dimensional Wiener process and Poisson random measure

In this paper, we investigate properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which depends on two parameters i.e., grid density n∈N and truncation dimension parameter M∈N, is of the order n−1/2+δ(M) such that δ(⋅) is positive and decreasing to 0. We derive a complexity model and provide proof for the complexity upper bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both n and M. The complexity is measured in terms of upper bound for mean-squared error and is compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation details are also reported.

An analytic method to investigate hemodynamics of the cardiovascular system: Biventricular system.

Previously, we found analytic solutions for single ventricular system based on the lumped parameter model (LPM). In this study, we generalized the method to biventricular system and derived its analytic solutions. LPM is just a set of differential equations, but it is difficult to solve due to time-varying ventricular elastance and high order. Mathematically, there exist no elementary solutions for time-varying equations. It turns out that instead of differential equations, according to volume conservation, a set of algebraic equations can be carried out. The solutions of the set of equations are just physiological states at end of systolic and diastolic phases such as end systolic/diastolic pressure/volume of left ventricle. As a preliminary application, the method is utilized to deduce the hemodynamic effects of VA ECMO. Left ventricular (LV) distension, a serious complication of VA ECMO, is usually attributed to factors such as increased afterload, inadequate LV unloading, reduced myocardial contractility or aortic valve regurgitation (AR), bronchial and Thebesian return in the absence of aortic valve (AoV) opening. Among these, reduced contractility and AR are strongly associated with LV distension. However, in the absence of reduced contractility or AR, it is less clear whether increased afterload or inadequate LV unloading alone can cause LV distension. This leads to the critical question: under what conditions does LV distension occur in the absence of reduced contractility or AR? The analytic formulas derived in this study give conditions for LV distension. Furthermore, the results show that the analytic hemodynamics are coincident with simulated results.

Nonlinear Poisson-Boltzmann solutions for charged parallel plates: When opposite charges repel.

I present an exact solution of the Poisson-Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, Lc,1 < Lc,2. For separations less than Lc,1, the force between plates is repulsive. It switches to attractive at Lc,1, but with the electric potential having the same signon both plates. For L > Lc,2, the force remains attractive, and the potential at the plates has the same signas the charge on each plate. I also describe charge regulation, determined by pKa, and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. The implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.

Impact of Multi-Scattered LiDAR Returns in Fog.

In the context of autonomous driving, the augmentation of existing data through simulations provides an elegant solution to the challenge of capturing the full range of adverse weather conditions in training datasets. However, existing physics-based augmentation models typically rely on single scattering approximations to predict light propagation under unfavorable conditions, such as fog. This can prevent the reproduction of important signal characteristics encountered in a real-world environment. Consequently, in this work, Monte Carlo simulations are employed to assess the relevance of multiple-scattered light to the detected LiDAR signal in different types of fog, with scattering phase functions calculated from Mie theory considering real particle size distributions. Bidirectional path tracing is used within the self-developed GPU-accelerated Monte Carlo software to compensate for the unfavorable photon statistics associated with the limited detection aperture of the LiDAR geometry. To validate the Monte Carlo software, an analytical solution of the radiative transfer equation for the time-resolved radiance in terms of scattering orders is derived, thereby providing an explicit representation of the double-scattered contributions. The results of the simulations demonstrate that the shape of the detected signal can be significantly impacted by multiple-scattered light, depending on LiDAR geometry and visibility. In particular, double-scattered light can dominate the overall signal at low visibilities. This indicates that considering higher scattering orders is essential for improving AI-based perception models.

Existence and exponential stability in pth moment of non-autonomous stochastic integro-differential equations

This study is concerned with the existence and exponential stability of solutions of non-autonomous neutral stochastic integro-differential equations with delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic integro-differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences. Finally, we provide an example to illustrate the results.

Jump relation for the conormal derivative of parabolic single layer potential

We consider a single layer potential generated by the fundamental solution of a second-order parabolic equation with Dini continuous coefficients. The potential is considered in a semibounded spatial domain with non-smooth lateral boundary from the Dini–Hölder class. We establish a jump relation for the conormal derivative of this potential at the lateral boundary of the domain.

On the Qualitative Analysis of Solutions of Two Fractional Order Fractional Differential Equations

In applied sciences, besides the importance of obtaining the analytical solutions of differential equations with constant coefficients, the qualitative analysis of the solutions of such equations is also very important. Due to this importance, in this study, a qualitative analysis of the solutions of a delayed and constant coefficient fractal differential equation with more than one fractional derivative was performed. In the equation under consideration, the derivatives are the Riemann–Liouville fractional derivatives. In the proof of the obtained results, Laplace transform formulas of the Riemann–Liouville fractional derivative and some inequalities are used. We also provide some examples to check the accuracy of our results.

Existence of positive solutions for a class of p-Laplacian fractional differential equations with nonlocal boundary conditions

This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results.

Mixed localized waves and their interaction structures for a spatial discrete Hirota equation

Mixed localized wave solutions and interactions are of great significance in nonlinear physical systems. This paper aims to investigating the generalized (m,N-m)-fold Darboux transformation and the mixed localized wave solutions of a spatial discrete Hirota equation. First, we construct the generalized (m,N-m)-fold Darboux transformation for the spatial discrete Hirota equation, which can produce the interactions between the breathers, degenerate breathers and rogue waves. For the Darboux transformation formula, we discuss the above order-1,2,3 localized wave solutions, as well as their dynamics by choosing the number of m = 1. We plot some specific examples such as the spatial (time)-periodic breather, second-order and third-order degenerate breathers and higher-order rogue waves with novel patterns. Furthermore, when m > 1, we give several kinds of mixed interaction solutions between the first-order rogue waves and first (second)-order (degenerate) breathers, between the first-order breather and second-order degenerate breathers, between second-order rogue waves and first-order breathers. At last, we also sum up the various mathematical features of the degenerate breathers and the mixed localized wave solutions.

Boundedness of pseudo-differential operators via coupled fractional Fourier transform

In this article, we obtained some fruitful results of the coupled fractional Fourier transform and its kernel. We defined pseudo-differential operators related to coupled fractional Fourier transform on Schwartz spaces and it is shown that their composition is again a pseudo-differential operator. It made further discussion on the composition of two pseudo-differential operators on Sobolev spaces and derived certain norm inequalities on it. Further, we have successfully applied some of the results of the coupled fractional Fourier transform to investigate the solution of nth-order linear non-homogeneous partial differential equation and wave equation. Lastly, we present some examples involving graphs and tables to illustrate the validity of our theoretical findings.

Study of a combined Kairat-II-X equation: Painlevé integrability, multiple kink, lump and other physical solutions

Purpose This study aims to explore a novel model that integrates the Kairat-II equation and Kairat-X equation (K-XE), denoted as the Kairat-II-X (K-II-X) equation. This model demonstrates the connections between the differential geometry of curves and the concept of equivalence. Design/methodology/approach The Painlevé analysis shows that the combined K-II-X equation retains the complete Painlevé integrability. Findings This study explores multiple soliton (solutions in the form of kink solutions with entirely new dispersion relations and phase shifts. Research limitations/implications Hirota’s bilinear technique is used to provide these novel solutions. Practical implications This study also provides a diverse range of solutions for the K-II-X equation, including kink, periodic and singular solutions. Social implications This study provides formal procedures for analyzing recently developed systems that investigate optical communications, plasma physics, oceans and seas, fluid mechanics and the differential geometry of curves, among other topics. Originality/value The study introduces a novel Painlevé integrable model that has been constructed and delivers valuable discoveries.

Klein-Gordon potential in characteristic coordinates

Abstract By the Klein-Gordon potential, we call a convolution-type integral with a kernel, which is the fundamental solution of the Klein-Gordon equation and also a solution of the Cauchy problem to the same equation. An interesting question having several important applications (in general) is what boundary condition can be imposed on the Klein-Gordon potential on the boundary of a given domain so that the Klein-Gordon equation with initial conditions complemented by this “transparent” boundary condition would have a unique solution within that domain still given by the Klein-Gordon potential. It amounts to finding the trace of the Klein-Gordon potential to the boundary of the given domain. In this article, we analyze this question and construct a novel initial boundary-value problem for the Klein-Gordon equation in characteristic coordinates.