Multidimensional Equations Research Articles

A Study of Nonlinear Riccati Equation and Its Applications to Multi-dimensional Nonlinear Evolution Equations

A Study of Nonlinear Riccati Equation and Its Applications to Multi-dimensional Nonlinear Evolution Equations

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform

Abstract This investigation explores the analytical solutions to the time-fractional multi-dimensional Navier–Stokes (NS) problem using advanced approaches, namely the Aboodh residual power series method and the Aboodh transform iteration method, within the context of the Caputo operator. The NS equation governs the motion of fluid flow and is essential in fluid dynamics, engineering, and atmospheric sciences. Given the equation’s extensive and diverse applicability across several disciplines, we are motivated to conduct a thorough analysis to understand the complex dynamics associated with the nonlinear events it describes. For this purpose, we effectively handle the challenges posed by fractional derivatives by utilizing the Aboodh approach. This will enable us to obtain accurate analytical approximations for the time fractional multi-dimensional NS equation. By conducting thorough analysis and computational simulations, we provide evidence of the efficiency and dependability of the suggested methodologies in accurately representing the dynamic behavior of fractional fluid flow systems. This work enhances our comprehension of the utilization of fractional calculus in fluid dynamics and provides valuable analytical instruments for examining intricate flow phenomena. Its interdisciplinary nature ensures that the findings are applicable to various scientific and engineering fields, making the research highly versatile and impactful.

Robust and accurate numerical framework for multi-dimensional fractional-order telegraph equations using Jacobi/Jacobi-Romanovski spectral technique

This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs.

Adaptive hyperbolic-cross-space mapped Jacobi method on unbounded domains with applications to solving multidimensional spatiotemporal integrodifferential equations

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

Correctness of the mixed problem for degenerate multidimensional elliptic-parabolic equations

The initial-boundary value problem (Dirichlet problem) for general elliptic-parabolic equations of second order was first posed by G. Fichera. Further investigation of this problem was carried out in the monograph by O.A. Oleinik and E.V. Radkevich and the works by V.N. Vragov. In these works, the authors examined mixed problems for degenerate multidimensional elliptic equations. The articles by S.A. Aldashev focused on the correctness (in the sense of uniqueness of solvability) of the Dirichlet problem in a cylindrical domain for multidimensional elliptic-parabolic equations. A mixed problem for these equations has not been studied. In this paper, the authors demonstrate the uniqueness of solvability and obtain an explicit representation of the classical solution of the mixed problem for degenerate multidimensional elliptic-parabolic equations. The proposed method allows reducing the problem under study to a mixed problem for a degenerate multidimensional elliptic equation examined by S.A. Aldashev.

Numerical solution of source identification multi-point problem of parabolic partial differential equation with Neumann type boundary condition

We study a source identification boundary value problem for a parabolic partial differential equation with multi-point Neumann type boundary condition. Stability estimates for the solution of the overdetermined mixed BVP for multi-dimensional parabolic equation were established. The first and second order of accuracy difference schemes for the approximate solution of this problem were proposed. Stability estimates for both difference schemes were obtained. The result of numerical illustration in test example was given.

On multidimensional exact solutions of the nonlinear diffusion equation of the pantograph type with variable delay

We consider a multidimensional pantograph-type nonlinear diffusion equation with a linearly increasing time delay and scaling with respect to spatial variables in the source (sink). It is proposed to construct exact solutions by the reduction method using two ansatzes with a quadratic dependence on spatial variables. The dependence of the solution on spatial variables is found from a system of algebraic equations, and the dependence on time is found from a system of ordinary differential equations with a linearly increasing delay of the argument. A number of examples of exact solutions are given, both radially symmetric and anisotropic with respect to spatial variables.

Optimistic value models of saddle point equilibrium control problems for uncertain jump systems

The saddle point equilibrium control problem is of great importance in differential game theory. In this paper, we investigated the optimistic value model for the saddle point equilibrium control problem based on multi-dimensional uncertain differential equations with jumps. The equilibrium equation for the proposed model is presented. Then a linear quadratic game optimistic value model and a bang-bang game optimistic value model are discussed, respectively. Finally, some applications are analysed by the results obtained in this paper.

The modelling error in multi-dimensional time-dependent solute transport models

Starting from full-dimensional models of solute transport, we derive and analyze multi-dimensional models of time-dependent convection, diffusion, and exchange in and around pulsating vascular and perivascular networks. These models are widely applicable for modelling transport in vascularized tissue, brain perivascular spaces, vascular plants and similar environments. We show the existence and uniqueness of solutions to both the full- and the multi-dimensional equations under suitable assumptions on the domain velocity. Moreover, we quantify the associated modelling errors by establishing a-priori estimates in evolving Bochner spaces. In particular, we show that the modelling error decreases with the characteristic vessel diameter and thus vanishes for infinitely slender vessels. Numerical tests in idealized geometries corroborate and extend upon our theoretical findings.

Resilience of weighted networks with dynamical behavior against multi-node removal.

In many real-world networks, interactions between nodes are weighted to reflect their strength, such as predator-prey interactions in the ecological network and passenger numbers in airline networks. These weighted networks are prone to cascading effects caused by minor perturbations, which can lead to catastrophic outcomes. This vulnerability highlights the importance of studying weighted network resilience to prevent system collapses. However, due to many variables and weight parameters coupled together, predicting the behavior of such a system governed by a multi-dimensional rate equation is challenging. To address this, we propose a dimension reduction technique that simplifies a multi-dimensional system into a one-dimensional state space. We applied this methodology to explore the impact of weights on the resilience of four dynamics whose weights are assigned by three weight assignment methods. The four dynamical systems are the biochemical dynamical system (B), the epidemic dynamical system (E), the regulatory dynamical system (R), and the birth-death dynamical system (BD). The results show that regardless of the weight distribution, for B, the weights are negatively correlated with the activities of the network, while for E, R, and BD, there is a positive correlation between the weights and the activities of the network. Interestingly, for B, R, and BD, the change in the weights of the system has little impact on the resilience of the system. However, for the E system, the greater the weights the more resilient the system. This study not only simplifies the complexity inherent in weighted networks but also enhances our understanding of their resilience and response to perturbations.

A semi-analytical solutions of the multi-dimensional time-fractional Klein-Gordon equations using residual power series method

This study presents a novel approach to getting a semi-analytical solution to the multi-dimensional time-fractional linear and nonlinear Klein–Gordon equations with appropriate initial conditions using the residual power series method. The time-fractional derivative (β) is used in the context of the Caputo approach. Some test examples of KGEs are considered to illustrate the validity and efficiency of the employed RPS method. The RPS solutions are compared with the exact solutions for β = 2 to ensure the method’s reliability and precision. The error bound and convergence analysis of the proposed method are also examined. The effects of the distinct values of fractional order β ∈ (1, 2] on the behavior of the proposed equations are also discussed.

A new shifted generalized Chebyshev approach for multi-dimensional sinh-Gordon equation

The numerical treatment of multi-dimensional non-linear sinh-Gordon equations is the focus of this paper. We numerically solve the (1 + 1) and (2 + 1) sinh-Gordon equations using two collocation algorithms. We select the set of basis functions as a set of generalized Chebyshev polynomials (CPs), which we express as orthogonal combinations of CPs. We develop and utilize some formulas related to these polynomials to propose our numerical algorithms. Specific values for the high-order derivatives of the basis functions serve in the derivation of the two presented algorithms. Additionally, we provide an estimation of the basis functions used in the convergence analysis study. We follow the two collocation algorithms to transform the sinh-Gordon equations into non-linear equation systems, which any suitable solver can handle. We provide some examples and comparisons to confirm the effectiveness of our presented algorithms.

Painlevé integrability and multiple soliton solutions for the extensions of the (modified) Korteweg-de Vries-type equations with second-order time-derivative

This work introduces two (3+1)-dimensional expansions of the Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. These extensions incorporate a second-order time-derivative term, similar to the Boussinesq equation. The Painlevé test is utilized to verify the integrability of each extended model. The bilinear form is employed to investigate the existence of multiple-soliton (MS) solutions for each system under consideration. Furthermore, we provide solutions in the form of lumps for the extended KdV equation. The multidimensional KdV-type equations surpass the standard KdV equation. However, they offer enhanced accuracy by representing a broader spectrum of nonlinear phenomena in plasma physics, fluid mechanics, tsunami phenomena, and other science disciplines. Furthermore, the aforementioned equations can be employed to analyze the characteristics of various acoustic waves (AW) in different plasma models, such as their amplitude, width, frequency, and dispersion, as well as the phase shifts after collisions.

Florence Bullying-Victimization Scales: Validation Study and Victimization Associations With Well-Being and Social Self-Efficacy

This study provides a thorough psychometric evaluation of construct and criterion validity and measurement invariance of the promising Florence Bullying-Victimization Scales (FBVS). A special focus was devoted to the concurrent criterion validity of the victimization scale with regard to well-being and social self-efficacy. Exploratory and confirmatory multidimensional item response theory and structural equation modeling were applied to cross-sectional data retrieved from 3rd to 6th-grade Czech primary school students ( N = 1795; 49% female; M age = 10.42, SD = 1.25). The results supported the use of unidimensional factor structure that demonstrated acceptable model fit and measurement invariance across genders and grades. Moderate to high correlations of the FBVS scores with bullying and victimization measured by the Olweus Bully/Victim Questionnaire and other instruments indicated very good convergent validity. Regarding criterion validity, higher victimization was associated with lower levels of well-being and social self-efficacy.

Hybrid technique for multi-dimensional fractional diffusion problems involving Caputo–Fabrizio derivative

In this study, a precise and analytical method, namely Shehu transform decomposition method (STDM), is applied to examine multi-dimensional fractional diffusion equations, which will describe density dynamics in a material undergoing diffusion. The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative. The proposed approach combines the Shehu transformation (ST) with the Adomian decomposition method (ADM), employing Adomain polynomials to handle nonlinear terms. Fixed point theorem has been used to prove the uniqueness and existence of the solution of governing differential equation. To validate the accuracy of proposed method, we present the analytical solutions of three applications of multi-dimensional fractional diffusion equations. Furthermore, we compute the graphical and numerical results using MATLAB to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem.

Pseudospectral analysis for multidimensional fractional Burgers equation based on Caputo fractional derivative

This study presents the Chebyshev pseudospectral approach in time and space to approximate a solution to the time-fractional multidimensional Burgers equation. The suggested approach utilizes Chebyshev–Gauss–Lobatto (CGL) points in both spatial and temporal directions. To figure out the fractional derivative matrix at CGL points, we use the Caputo fractional derivative formula. Further, the Chebyshev fractional derivative matrix is utilized to reduce the given problem in an algebraic system of equations. The numerical approach known as the Newton–Raphson is implemented to get the desired results for the system. Error analysis for the set of values of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ u $$\\end{document} is done for various model examples of fractional Burgers equations, where ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} represents the fractional order. The computed numerical results are in perfect agreement with the exact solutions.

Multidimensional Hyperbolic Equations with Nonlocal Potentials: Families of Global Solutions

In spaces of arbitrary dimensions, hyperbolic differential–difference equations with potentials undergoing translations along arbitrary spatial directions are investigated. The fundamental novelty of this study is that, unlike previous investigations, no restrictions are imposed on the symbol of the differential–difference operator contained in the equation. For the investigated equation, we succeed to explicitly construct a multiparametric family of classical global solutions.

Global smooth solutions of multidimensional compressible Euler equations with critical time depending damping and vorticity

The authors (Hou and Yin, 2017 [9]; Hou et al., 2018 [10]) proved that the 2D and 3D irrotational compressible Euler equations with critical time depending damping admit global classical solutions. In this paper, we will study this problem with non-zero vorticity and show the global existence of smooth solutions to the 2D and 3D compressible Euler equations with critical time depending damping. They used the Lorentz vector field and their method fails in the presence of the vorticity. The main ingredients in this paper are the auxiliary energies which will make up for the loss of the Lorentz vector field and the improved decay estimates of the energy which can be achieved by a new multiplier.

Emergence of the reproduction matrix in epidemic forecasting.

During the recent COVID-19 pandemic, the instantaneous reproduction number, R(t), has surged as a widely used measure to target public health interventions aiming at curbing the infection rate. In analogy with the basic reproduction number that arises from the linear stability analysis, R(t) is typically interpreted as a threshold parameter that separates exponential growth (R(t) > 1) from exponential decay (R(t) < 1). In real epidemics, however, the finite number of susceptibles, the stratification of the population (e.g. by age or vaccination state), and heterogeneous mixing lead to more complex epidemic courses. In the context of the multidimensional renewal equation, we generalize the scalar R(t) to a reproduction matrix, [Formula: see text], which details the epidemic state of the stratified population, and offers a concise epidemic forecasting scheme. First, the reproduction matrix is computed from the available incidence data (subject to some a priori assumptions), then it is projected into the future by a transfer functional to predict the epidemic course. We demonstrate that this simple scheme allows realistic and accurate epidemic trajectories both in synthetic test cases and with reported incidence data from the COVID-19 pandemic. Accounting for the full heterogeneity and nonlinearity of the infection process, the reproduction matrix improves the prediction of the infection peak. In contrast, the scalar reproduction number overestimates the possibility of sustaining the initial infection rate and leads to an overshoot in the incidence peak. Besides its simplicity, the devised forecasting scheme offers rich flexibility to be generalized to time-dependent mitigation measures, contact rate, infectivity and vaccine protection.