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AbstractWe examine an SIRS reaction-diffusion model with local dispersal and spatial heterogeneity to study COVID-19 dynamics. Using the operator semigroup approach, we establish the existence of disease-free equilibrium (DFE) and endemic equilibrium (EE), and derive the basic reproduction number, R0. Simulations show that without dispersal, reinfection and limited medical resources problems can cause a plateau in cases. Dispersal and spatial heterogeneity intensify localised outbreaks, while integrated control strategies (vaccination and treatment) effectively reduce infection numbers and epidemic duration. The possibility of reinfection demonstrates the need for adaptable health measures. These insights can guide optimised control strategies for enhanced public health preparedness. [ABSTRACT FROM AUTHOR]

Under study is some two-dimensional second order nonlinear hyperbolic equation with variable coefficients. The left-hand side of the equation contains quadratic nonlinearities with the unknown function and its derivatives. We consider the linear multiplicative transformations of the unknown function which preserve the form of the initial equation. By analogy with linear equations, the Laplace invariants are determined as the invariants of such transformation. We find the expressions of the Laplace invariants which use the coefficients of the equation and their first derivatives. Also, we consider the general case and the case that some coefficients of the equation equal to zero as well as prove the main theorem about Laplace invariants. By the theorem, two nonlinear hyperbolic equations under study can be connected by using a linear multiplicative transformation if only if the Laplace invariants for both equations are the same. We find the equivalent systems of the first order equations containing the Laplace invariants, in the general case and in the case that some coefficients of the equation vanish. We demonstrate that a solution to the initial equation can be received in quadratures under some additional conditions on the coefficients and the Laplace invariants. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the ϕ 6 model in dimension 1 + 1. We construct a sequence of approximate solutions (ϕ k (v , t , x)) k ∈ N ⩾ 2 for this model to understand the effects of the collision in the movement of each soliton during a large time interval. The construction uses a new asymptotic method which is not only restricted to the ϕ 6 model. [ABSTRACT FROM AUTHOR]

Cancer invasion and migration play a pivotal role in tumor malignancy, which is a major cause of most cancer deaths. Rotating magnetic field (RMF), one of the typical dynamic magnetic fields, can exert substantial mechanical influence on cells. However, studying the effects of RMF on cell is challenging due to its complex parameters, such as variation of magnetic field intensity and direction. Here, we developed a systematic simulation method to explore the influence of RMF on tumor invasion and migration, including a finite element method (FEM) model and a cell-based hybrid numerical model. Coupling with the data of magnetic field from FEM, the cell-based hybrid numerical model was established to simulate the tumor cell invasion and migration. This model employed partial differential equations (PDEs) and finite difference method to depict cellular activities and solve these equations in a discrete system. PDEs were used to depict cell activities, and finite difference method was used to solve the equations in discrete system. As a result, this study provides valuable insights into the potential applications of RMF in tumor treatment, and a series of in vitro experiments were performed to verify the simulation results, demonstrating the model's reliability and its capacity to predict experimental outcomes and identify pertinent factors. Furthermore, these findings shed new light on the mechanical and chemical interplay between cells and the ECM, offering new insights and providing a novel foundation for both experimental and theoretical advancements in tumor treatment by using RMF. [ABSTRACT FROM AUTHOR]

With the goal to support effective water resource management, water quality models have gained popularity as tools for evaluating the distributions of pollutants and sediments. This work focuses on the application of the numerical solution of an advection-dispersion-reaction (ADR) water quality model for rivers and streams to a major Philippine waterway, the Pasig River. The water quality constituent is described by a system of reaction and advection-dispersion-reaction equations. The model and method are based on a previously used strategy where Guass-Jordan decomposition is applied to the matrix system and the resulting conservative form of the model is solved numerically using the fully implicit scheme and finite element method. The methodology is demonstrated by a case study in Pasig River involving the concentrations of total dissolved solids (TDS) obtained from the Department of Environment and Natural Resources (DENR) through the Pasig River Unified Monitoring Stations (PRUMS) report. Sensitivity analysis and parameter estimation are also applied to the model to assess which parameters influence the model output the most. [ABSTRACT FROM AUTHOR]

In this paper, some symmetrized two-scale finite element methods are proposed for a class of partial differential equations with symmetric solutions. With these methods, the finite element approximation on a fine tensor-product grid is reduced to the finite element approximations on a much coarser grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximations still maintain an asymptotically optimal accuracy. By symmetrized two-scale finite element methods, the computational cost can be reduced further by a factor of 푑 approximately compared with two-scale finite element methods when Ω = (0 , 1) d . Consequently, symmetrized two-scale finite element methods reduce computational cost significantly. [ABSTRACT FROM AUTHOR]

In this present study, we reduce the fractional reaction-diffusion equation to a traditional differential equation using the fractional complex transformation and consider the Landau Lifshitz (LLG) equation. Moreover, by using the generalized exponential rational function method and Kudryashov's method respectively we extract new exact and solitary wave solutions for these equations. Some plots of some presented new solutions are represented to exhibit wave characteristics. All results in this paper are essential to understand the physical meaning and behavior of the investigated equation that sheds light on the importance of investigating various nonlinear wave phenomena in mathematical physics. [ABSTRACT FROM AUTHOR]

This paper addresses the robust vibration control problem for a gantry crane system which is governed by a partial differential equation. The control objectives are to transport the cargo from the initial position to the desired position and suppress the vibration of the cable and payload in the presence of boundary disturbance. To achieve the objectives, a sliding-mode controller is proposed by repressing the boundary disturbance. By using the operator semigroup theory, the well-posedness of the closed-loop system is guaranteed. The asymptotic stability of the closed-loop system is proven by using the extended LaSalle's invariance principle. Both numerical simulations and experiments are provided to illustrate the effectiveness of the proposed boundary control method. [ABSTRACT FROM AUTHOR]

Machine learning methods are progressively investigated for a large amount of applications. Recently, the solution of partial differential equations (PDE) describing problems in elastostatics came into focus. The current work investigates two neural network-based classes of methods for their solution, namely the neural finite element method (FEM) and neural operator methods. The analysis of these approaches is carried out by means of numerical experiments with linear and nonlinear material behavior where the conventional FEM serves as a benchmark. The formulation of neural FEM allows for elegant integration of finite deformation hyperelasticity at medium training effort. Here, training data are replaced by the evaluation of the equilibrium PDE at sample points. In contrast, most neural operator methods require expensive training with large data sets, but then allow for solving multiple boundary value problems with the same machine learning model. For the comparative analysis, the maximal relative error values over the whole domain and over all components of the strain tensor are evaluated as accuracy measure. The current state of research shows that none of the methods investigated reaches the accuracy and computational performance of the conventional FEM. In many standard applications, the FEM achieves an accuracy of 10 - 6 , whereas the numerical tests in the present work report a relative error of order of magnitude of 10 - 4 for the neural FEM and 10 - 2 to 10 - 3 for neural operator methods. [ABSTRACT FROM AUTHOR]

To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network defined on the whole problem domain. In such a partitioned neural network approach, suitable interface conditions or subdomain boundary conditions are combined to obtain a convergent approximate solution. However, there has been no rigorous study on the convergence and parallel computing enhancement on the partitioned neural network approach. In this paper, iterative algorithms are proposed to enhance parallel computation performance in the partitioned neural network approximation. Our iterative algorithms are based on classical additive Schwarz domain decomposition methods. For the proposed iterative algorithms, their convergence is analyzed under an error assumption on the local and coarse neural network solutions. Numerical results are also included to show the performance of the proposed iterative algorithms. [ABSTRACT FROM AUTHOR]

The aerodynamic drag of high-speed trains has a negative effect on their running stability and energy efficiency. Since the shape of the high-speed train head closely influences its surrounding airflow, optimizing the head shape is the primary way to reduce the aerodynamic drag. However, existing optimization methods have limitations in parametrically describing the train head with enough details and fewer parameters. In this paper, we propose a novel parametric modeling method based on the approximate analytical partial differential equation (PDE) for the aerodynamic shape optimization of high-speed train heads. With this method, the detailed shape of the train head is controlled by four design parameters. To enhance the optimization efficiency, a two-stage multilayer perceptron (MLP) surrogate model is proposed to predict the aerodynamic drag coefficients of the high-speed train, and a classic genetic algorithm (GA) is adopted to optimize the total drag coefficient and generate the train head shape with good aerodynamic performance. The effectiveness of the proposed method is demonstrated through several comparison experiments. [ABSTRACT FROM AUTHOR]

This paper addresses the vibration reduction and disturbance attenuation problems of a flexible arm, which is modelled as a fourth-order partial differential equation. The main control objectives lie in reducing the influence of the disturbance, steering the flexible arm to the desired angular position and suppressing the vibration, simultaneously. To achieve the objectives, firstly, two different disturbance observers are designed. Secondly, we construct boundary controllers based on the two designed disturbance observers and boundary measurable signals. Then, with the two disturbance observers, the boundary disturbance compensation is achieved, and with the designed boundary controllers, the uniform boundedness of the flexible arm is guaranteed. Moreover, the well-posedness of the closed-loop system under the designed boundary controllers is discussed by means of the operator semigroup theory. Finally, the effectiveness and advantages of the proposed controllers are demonstrated through simulations and physical experiments. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Education & Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota's results on the author's research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differential equations. [ABSTRACT FROM AUTHOR]

This work studies the complete lifts of Kenmotsu manifolds associated with the generalized Tanaka-Webster connection (GTWC) in the tangent bundle. Using the GTWC, this study explores the complete lifts of various curvature tensors and geometric structures from Kenmotsu manifolds to their tangent bundles. Specifically, it examines the complete lifts of Ricci semi-symmetry, the projective curvature tensor, $ \Phi $-projectively semi-symmetric structures, the conharmonic curvature tensor, the concircular curvature tensor, and the Weyl conformal curvature tensor. Additionally, the research delves into the complete lifts of Ricci solitons on Kenmotsu manifolds with the GTWC within the tangent bundle framework, providing new insights into their geometric properties and symmetries in the lifted space. The data on the complete lifts of the Ricci soliton in Kenmotsu manifolds associated with the GTWC in the tangent bundle are also investigated. An example of the complete lifts of a $ 5 $-dimensional Kenmotsu manifold is also included.

Abstract This paper presents a subdivision collocation algorithm for numerically solving the heat conduction equation with non-uniform thermal diffusivity, considering both initial and boundary conditions. The algorithm involves transforming the differential form of the heat conduction equation into a system of equations and discretizing the time variable using the finite difference formula. The numerical solution of the system of heat conduction equations is then obtained. The feasibility of the algorithm is verified through theoretical and numerical analyses. Additionally, numerical and graphical representations of the obtained numerical solutions are provided, along with a comparison to existing methods. The results demonstrate that our proposed method outperforms the existing methods in terms of accuracy.

This paper introduces a novel method that leverages the capability of the Adam Optimization- Back Propagation model to adaptively select shape parameters in the Multiple-Quadratic Radial Basis Function (MQRBF). This approach has been effectively combined with the Finite Difference (FD) method to generate high-precision solutions for complex Partial Differential Equations (PDEs). The careful selection of shape parameters in MQRBF is crucial to ensure the accuracy of the MQRBF-FD in addressing complex PDE scenarios. We have improved the Random Walk Optimization algorithm and integrated it with Fourier theory and deep learning techniques, establishing a comprehensive framework for adaptive optimization of shape parameters. This significantly enhances the adaptability and accuracy of the MQRBF-FD. A wide range of numerical experiments, including analyses of heat conduction in non-uniform materials and dye transport in fluid channels, highlight the exceptional accuracy, computational efficiency, and versatility of our method. [ABSTRACT FROM AUTHOR]

Fractional partial differential equations (FPDEs) have gained significant attention in various scientific and engineering fields due to their ability to describe complex phenomena with memory and long-range interactions. Solving FPDEs analytically can be challenging, leading to a growing need for efficient numerical methods. This review article presents the recent analytical and numerical methods for solving FPDEs, where the fractional derivatives are assumed in Riemann-Liouville’s sense, Caputo’s sense, Atangana-Baleanu’s sense, and others. The primary objective of this study is to provide an overview of numerical techniques commonly used for FPDEs, focusing on appropriate choices of fractional derivatives and initial conditions. This article also briefly illustrates some FPDEs with exact solutions. It highlights various approaches utilized for solving these equations analytically and numerically, considering different fractional derivative concepts. The presented methods aim to expand the scope of analytical and numerical solutions available for time-FPDEs and improve the accuracy and efficiency of the techniques employed.

Brain tumors are abnormal cell growths inside the skull that damage brain cells needed for brain function. The complex structure of the human brain makes it challenging to identify and categorize brain tumors. Nevertheless, segmenting brain tumors in MRI images is difficult due to the wide variation in tumor size, location, and intensity. Specifically, we propose a method with four modules: (i) preprocessing; (ii) decomposition; (iii) segmentation; and (iv) classification—that together overcome these difficulties. Here, we first remove the noise from the MRI brain image using a partial differential equation. Those pre-processed images are fed into the contourlet transform which works on the principle of multiscale decomposition of images. The contourlet transform employs a double filter bank structure comprised of the Laplacian pyramid and a directional filter bank to obtain a sparse representation of the smooth contour of an image. These extracted bands are segmented using a novel Possibilistic Fuzzy C-Means clustering algorithm. Brain tissue portions are finally classified into white matter, grey matter, cerebrospinal fluid, edema, and tumor tissues using an Optimized Support Vector Machine whose parameters are optimally chosen using an Opposition-based Grey Wolf Optimization algorithm. The BraTS2021 and Figshare datasets were used to evaluate the proposed method in terms of sensitivity, specificity, accuracy, PPV, NPV, FPR, and FNR. According to the experimental findings, the proposed methodology is superior to the conventional methods. Overall, the analysis demonstrates that the proposed method is more effective than the alternatives. [ABSTRACT FROM AUTHOR]

Partial differential equation-based methods have been widely applied in image restoration. The anisotropic diffusion model has a good noise removal capability without affecting significant edges. However, existing anisotropic diffusion-based models closely depend on the diffusion coefficient function and threshold parameter. This paper proposes a new weighted anisotropic diffusion coefficient model with multiple scales, and it has a higher speed of closing to X-axis and exploits adaptive threshold parameters. Meanwhile, the proposed algorithm is verified to be suitable for multiple types of noise. Numerical metrics and visual comparison of simulation experiments show the proposed model has significant superiority in edge-preserving and staircase artifacts reducing over the existing anisotropic diffusion-based techniques. • We find the weighted anisotropic diffusion coefficient function with high convergence speed. • The adaptive threshold parameter helps keep more details in restored images. • Multi-scale feature map fusing can reduce staircase artifacts along edges. • The performance of the proposed method is promising for real natural and medical images. [ABSTRACT FROM AUTHOR]

We prove the existence of solutions to the boundary value problem for a system of five nonlinear second order partial differential equations, with given nonlinear boundary conditions, describing the equilibrium state of elastic shallow inhomogeneous anisotropic shells with free edges within the framework of Timoshenko's shear model, referred to arbitrary curvilinear coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle. [ABSTRACT FROM AUTHOR]

Recent advancements in the integration of artificial intelligence (AI) and machine learning (ML) with physical sciences have led to significant progress in addressing complex phenomena governed by nonlinear partial differential equations (PDEs). This paper explores the application of novel operator learning methodologies to unravel the intricate dynamics of flame instability, particularly focusing on hybrid instabilities arising from the coexistence of Darrieus–Landau (DL) and Diffusive–Thermal (DT) mechanisms. Training datasets encompass a wide range of parameter configurations, enabling the learning of parametric solution advancement operators using techniques such as parametric Fourier Neural Operator (pFNO) and parametric convolutional neural networks (pCNNs). Results demonstrate the efficacy of these methods in accurately predicting short-term and long-term flame evolution across diverse parameter regimes, capturing the characteristic behaviors of pure and blended instabilities. Comparative analyses reveal pFNO as the most accurate model for learning short-term solutions, while all models exhibit robust performance in capturing the nuanced dynamics of flame evolution. This research contributes to the development of robust modeling frameworks for understanding and controlling complex physical processes governed by nonlinear PDEs. [ABSTRACT FROM AUTHOR]

In this work, the main discussion centres on avoiding the use of triangle inequality while proving Cauchy sequence to establish best proximity point theorems for single valued as well as multivalued non-self mappings. We prove such best proximity point theorems in the setting of non-triangular metric spaces and elaborate through examples. In this process host of the existing best proximity results are generalized and improved. To arouse further interest in the subject, we connect this work in solving a specific type of partial differential equation problem. [ABSTRACT FROM AUTHOR]

The present study focuses on numerical simulation of generalized FitzHugh-Nagumo equation employing shifted Chebyshev spectral collocation method (SCSCM). The collocation method makes use of shifted Chebyshev polynomials for approximation of spatial variable and its derivatives, whereas, Chebyshev-Gauss-Lobatto points are employed for collocation purpose. The approximation of FitzHugh-Nagumo equation gives rise to a system of nonlinear ordinary differential equations (ODEs). The solution of this system of ODEs has been obtained by using Runge-Kutta scheme of order 4. The obtained numerical solutions are shown in graphical and tabular form. The convergence of SCSCM has been demonstrated as well. Further, to verify the efficiency and accuracy of the present method, the absolute, maximum absolute, root mean square and relative errors have been calculated for some examples of FitzHugh-Nagumo equation. The comparison of present numerical solutions with exact and approximate solutions obtained by other methods reveals that present method provides better accuracy. [ABSTRACT FROM AUTHOR]

With the rapid advancement of quantum computers in the past few years, there is ongoing development of algorithms aimed at solving problems that are difficult to tackle with classical computers. A pertinent instance of this is the resolution of partial differential equations (PDEs), where a current trend involves the exploration of variational quantum algorithms (VQAs) tailored to efficiently function on the noisy intermediate-scale quantum (NISQ) devices. Recently, VQAs for solving the Poisson equation have been proposed, and these algorithms require highly entangled quantum states or specific types of qubit entanglement to compute the expectation value of the Laplace operator. Implementing such requirements on NISQ devices poses a significant challenge. To overcome this problem, we propose a new method for representing the Laplace operator in the finite difference formulation. Since the quantum circuits introduced for evaluating the expectation value of the Laplace operator through proposed method do not require processes that degrade the fidelity of computation, such as swap operations or generation of highly entangled states, they can be easily implemented on NISQ devices. In the regime of quantum supremacy (the number of qubits is approximately 50), our proposed approach necessitates approximately one-third fewer CNOT operations compared to conventional methods. To assess the effectiveness of the proposed method, we conduct computations for finding the eigenvalues of the Helmholtz equation and solving the Poisson equation on cloud-based quantum hardware. We calculate the fidelity of quantum states required for each method through quantum tomography and also estimate the fidelity in the quantum supremacy regime. We believe that the proposed method can be applied to other PDEs having the Laplace operator and greatly assists in solving them. [ABSTRACT FROM AUTHOR]

We study multidimensional nonautonomous evolution Monge–Ampère type equations. The left-hand side of such equation contains the first time derivative with the coefficient depending on time, space variables, and an unknown function. The right-hand side of the equation is the determinant of a Hessian matrix. We find the solutions by additive and multiplicative separation of variables and show that the representability of the coefficient of the time derivative as the product of functions of time and space variables is a sufficient condition for the existence of such solutions. In the case that the coefficient of the time derivative is the inverse function to a linear combination of space variables with coefficients depending on time, we also give solutions in the form of the quadratic polynomials in space variables. Also, we obtain the solution set in the form of the linear combination of functions of space variables with time depending coefficients. We consider some reductions of the equation to ODEs in the cases that the unknown function depends on the sum of functions of space variables (in particular, the sum of their squares) and a function of the time; in this case we use the functional separation of variables. Some reductions are also found of the given equation to PDEs of lower dimension. In particular, we find the solutions in the form of function of the time and the sum of squares of space variables as well as the solutions in the form of the sum of such functions. [ABSTRACT FROM AUTHOR]

Recently, the physics-informed neural network (PINN) has attracted much attention in solving partial differential equations (PDEs). The success is due to the strong generalization ability of the neural network (NN), which is supported by the universal approximation theorem, and its mesh-free implementation. In this paper, we propose a multi-head NN enriched PINN (MultiPINN) for solving differential equations. The trial function is built based on the radial basis function (RBF)-interpolation, which makes NN training parameters partially interpre. The loss function is constructed by embedding the physics information of differential equations and boundary conditions. Then the parameters in MultiPINN are trained using the ADAM optimizer. A significant feature of MultiPINN is that it combines the traditional RBF interpolation method with machine learning (ML) techniques. The ML technique is employed to learn the basis feature enrichment that provides global information. The multi-head mechanism is used so that each node has multiple bases, which can improve the accuracy of the MultiPINN solution. Two ordinary differential equations and three partial differential equations, i.e. the convection equation, the Burgers equation, and the Poisson equation, are used in the numerical experiments. The experimental outcomes demonstrate that MultiPINN produces solutions consistent with both analytical solutions and solutions obtained through traditional numerical methods. Additionally, MultiPINN shows robustness and adaptability over the other NN-based methods in the implementations. [ABSTRACT FROM AUTHOR]

In this paper, boundary value problems in semi-bounded domains are considered for a fourth-order equation with multiple characteristics. The uniqueness of the solution is proven by the method of energy integrals. The existence of solutions is proved by the method of separation of variables. The solutions are constructed explicitly in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is justified. [ABSTRACT FROM AUTHOR]

A class of solved with respect to the th order derivative evolution linear equations in Banach spaces on the real axis is considered. The equation is not endowed by initial conditions. A theorem on criterion in terms of the Fourier transform for analytic in a containing bisectorial region functions with values in a Banach space is proved. On the basis of this criterion we define a class of bisectorial operators. For the equation with a bisectorial operator, it is proved a unique solution existence. Moreover, it is shown that the unique solution has the form of the convolution of the inverse Fourier transform for and of the right-hand side of the equation. The obtained general results are used in the study of a boundary value problem to a class of partial differential equations. [ABSTRACT FROM AUTHOR]

The paper considers an optimal control problem in which the plant is described by a linear functional equation in a Hilbert space and a control action is a change of the space. Sufficient conditions for the existence of a solution are obtained. The results are generalized to the case in which the plant is described by a linear variational inequality. [ABSTRACT FROM AUTHOR]

A crucial aspect of contemporary image processing systems is image denoising. The anisotropic diffusion function is a feature of the partial differential equation employed for the purpose of noise reduction and the preservation of image characteristics such as edges. A new tangent sigmoid diffusion coefficient and a new adaptive threshold parameter have been proposed in this work, which leads to faster convergence. In comparison to traditional anisotropic diffusion model techniques, the proposed technique performs admirably. As evidenced by the results, which demonstrate that the new anisotropic diffusion technique is not only capable of efficiently removing noise, but also of maintaining content in the denoised image. The performance of the proposed method is evaluated using various metrics, including peak signal-to-noise ratio, convergence rate, structural similarity index, time complexity, and space complexity. When comparing the proposed approach to previous methods, it is evident that the proposed method outperforms in various aspects. These include a higher convergence rate (− 0.1278), a greater peak signal-to-noise ratio (37.9827 dB), a higher structural similarity index (0.97432), a lower time complexity (5.72 s), and a smaller space complexity (15.6 KB). [ABSTRACT FROM AUTHOR]

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations. [ABSTRACT FROM AUTHOR]

Uncertain partial differential equation (UPDE) was introduced in literature. But the solution of a UPDE was not defined well. In this article, we will rigorously give a suitable concept of a UPDE and define its solution by an integral equation. Then, some examples are given to show the rationality of the definition. Uncertain heat conduction equation is presented as an application of UPDE. For those UPDEs having no analytic solutions, α -path method is introduced to obtain the inverse uncertainty distributions of solutions to UPDEs. [ABSTRACT FROM AUTHOR]

Contemporary quantum computers open up novel possibilities for tackling intricate problems, encompassing quantum system modeling and solving partial differential equations (PDEs). This paper explores the optimization of quantum algorithms aimed at resolving PDEs, presenting a significant challenge within the realm of computational science. The work delves into the application of the Variational Quantum Eigensolver (VQE) for addressing equations such as Poisson’s equation. It employs a Hamiltonian constructed using a modified Feynman-Kitaev formalism for a VQE, which represents a quantum system and encapsulates information pertaining to the classical system. By optimizing the parameters of the quantum circuit that implements this Hamiltonian, it becomes feasible to achieve minimization, which corresponds to the solution of the original classical system. The modification optimizes quantum circuits by minimizing the cost function associated with the VQE. The efficacy of this approach is demonstrated through the illustrative example of solving the Poisson equation. The prospects for its application to the integration of more generalized PDEs are discussed in detail. This study provides an in-depth analysis of the potential advantages of quantum algorithms in the domain of numerical solutions for the Poisson equation and emphasizes the significance of continued research in this direction. By leveraging quantum computing capabilities, the development of more efficient methodologies for solving these equations is possible, which could significantly transform current computational practices. The findings of this work underscore not only the practical advantages but also the transformative potential of quantum computing in addressing complex PDEs. Moreover, the results obtained highlight the critical need for ongoing research to refine these techniques and extend their applicability to a broader class of PDEs, ultimately paving the way for advancements in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

Susceptibility of airborne ultrasonic power to augment heat and mass transfer during hot air dehydration of peppermint leaves was investigated in the present study. To predict the moisture removal curves, a unique non-equilibrium mathematical model was developed. For the samples dried at temperatures of 40‒70 °C and the power intensities of 0‒104 kW m−3, the diffusion of moisture inside the leaves and coefficients for of mass and heat transfer varied from 0.601 × 10–4 to 5.937 × 10–4 s−1, 4.693 × 10–4 to 7.975 × 10–4 m s−1 and 49.2 to 78.1 W m−2 K−1, respectively. In general, at the process temperatures up to 60 °C, all the studied transfer parameters were augmented in the presence of ultrasonic power. [ABSTRACT FROM AUTHOR]

We study the copolynomials, i.e., K-linear mappings from the ring of polynomials K[x] into the commutative ring K. With the help of the Cauchy–Stieltjes transform of a copolynomial, we introduce and examine a multiplication of copolynomials. We investigate the Cauchy problem related to the nonlinear partial differential equation ∂u∂t=aum0(∂u∂x)m1(∂2u∂x2)m2(∂3u∂x3)m3, m0,m1,m2,m3∈ℕ0, ∑j=03mj>0, a∈K in the ring of copolynomials. To find a solution, we use the series of powers of the δ-function. As examples, we consider the Cauchy problem with the Euler–Hopf equation ∂u∂t+u∂u∂x=0, for a Hamilton–Jacobi type equation ∂u∂t=(∂u∂x)2, and for the Harry Dym equation ∂u∂t=u3∂3u∂x3.

IntroductionCurrent demographic trends predict continuously growing numbers of individuals reliant on care, which has to be accounted for in future planning of long-term care-resources. The projection of developments becomes especially necessary in order to enable healthcare systems to cope with this future burden and to implement suitable strategies to deal with the demand of long-term care. This study aimed to project the prevalence of long-term care and the number of care-dependent people in Germany until 2050.MethodsWe used the illness-death model to project the future prevalence of long-term care in Germany until 2050 considering eight different scenarios. Therefore, transition rates (incidence rate and mortality rates) describing the illness-death model are needed, which have been studied recently. Absolute numbers of people in need for long-term care were calculated based to the 15th population projection of the Federal Statistical Office.ResultsNumbers of people in need for long-term care will increase by at least 12%, namely 5.6 million people, in the period of 2021 until 2050. Assuming an annual incidence-increase of 2% from 2021 to 2050 the number of care-dependent individuals could potentially rise up to 14 million (+180%).ConclusionOur projections indicated a substantial rise in the number of care-dependent individuals. This is expected to lead to raising economic challenges as well as a stronger demand for healthcare and nursing personnel.

With the goal to support effective water resource management, water quality models have gained popularity as tools for evaluating the distributions of pollutants and sediments. This work focuses on the application of the numerical solution of an advection-dispersion-reaction (ADR) water quality model for rivers and streams to a major Philippine waterway, the Pasig River. The water quality constituent is described by a system of reaction and advection-dispersion-reaction equations. The model and method are based on a previously used strategy where Guass-Jordan decomposition is applied to the matrix system and the resulting conservative form of the model is solved numerically using the fully implicit scheme and finite element method. The methodology is demonstrated by a case study in Pasig River involving the concentrations of total dissolved solids (TDS) obtained from the Department of Environment and Natural Resources (DENR) through the Pasig River Unified Monitoring Stations (PRUMS) report. Sensitivity analysis and parameter estimation are also applied to the model to assess which parameters influence the model output the most.

This work focuses on the non-linear dynamics of unsteady tri-hybrid nanofluids inside a thermally radiative, expandable or contractible cylinder, which has been a consistent subject of interest for contemporary researchers. An advanced model is formulated to address the unsteady external flowing of a conductive 50 %:50 % by volume aqueous solution of ethylene glycol-based ternary hybrid nanofluid (THNF) over a linearly stretched cylindrical structure, considering the significant applications of an anti-freeze agent in industrial, construction, nuclear, and mechanical domains. The proposed tri-hybrid nanomaterials consist of cobalt magnetite, titania, and magnesium oxide nanoparticles. The heat transfer rate is analyzed concerning viscous dissipation, Ohmic heating, and convective boundary conditions. This research examines the effects of magnetic and induced electric fields. The non-dimensional similarity model for the controlling partial differential system is derived using appropriate transformations and solved by the homotopy analysis technique (HAM) to get series solutions. In contrast to porosity and magnetic characteristics, the flow rate increases with increasing unsteadiness and electric factors. Close to the surface, the Nusselt number rises with an increasing unsteadiness parameter, but skin frictions diminishes. THNF has been shown to significantly enhance the cooling of cylindrical conduits and mechanical antifreeze agents.