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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]

The mathematical modeling of a combination of non-thermal technologies for E. coli inactivation is of great interest for describing the dynamic behavior of microorganisms in food, with the goal of process control, optimization, and prediction. This research focused on the design and implementation of a mathematical model to predict the effect of power ultrasound (US), high-pressure processing (HPP), and the combination of both non-thermal technologies on the inactivation kinetics of E. coli (DSM682) inoculated in orange juice. Samples were processed by US, HPP, and a combination of both technologies at varying process parameters, and a mathematical model for microbial inactivation was developed using a System Dynamics approach. The results showed that the combination of these technologies exhibited a synergistic effect, resulting in no detectable colony-forming units per mL of juice. The developed model accurately predicted the inactivation of E. coli following the combination of these technologies (R2 = 0.82) and can be used to predict microbial load reduction or optimize it based on process parameters. Additionally, combining both techniques offers a promising approach for extending the shelf life of fresh juices using non-thermal stabilization technology. [ABSTRACT FROM AUTHOR]

With the rapid advancement of deep learning, computer-aided diagnosis and treatment have become crucial in medicine. UNet is a widely used architecture for medical image segmentation, and various methods for improving UNet have been extensively explored. One popular approach is incorporating transformers, though their quadratic computational complexity poses challenges. Recently, State-Space Models (SSMs), exemplified by Mamba, have gained significant attention as a promising alternative due to their linear computational complexity. Another approach, neural memory Ordinary Differential Equations (nmODEs), exhibits similar principles and achieves good results. In this paper, we explore the respective strengths and weaknesses of nmODEs and SSMs and propose a novel architecture, the nmSSM decoder, which combines the advantages of both approaches. This architecture possesses powerful nonlinear representation capabilities while retaining the ability to preserve input and process global information. We construct nmSSM-UNet using the nmSSM decoder and conduct comprehensive experiments on the PH2, ISIC2018, and BU-COCO datasets to validate its effectiveness in medical image segmentation. The results demonstrate the promising application value of nmSSM-UNet. Additionally, we conducted ablation experiments to verify the effectiveness of our proposed improvements on SSMs and nmODEs. [ABSTRACT FROM AUTHOR]

The authors develop and substantiate an algorithm for finding an ordinary differential equation of minimum order with polynomial coefficients over the field of rational numbers, whose solutions are a given system of polynomials (in particular, a system of modified Laguerre–Cayley polynomials). [ABSTRACT FROM AUTHOR]

Simple Summary: Immune checkpoint inhibitors (ICIs) are cancer immunotherapeutics that reinvigorate immune cells' ability to attack tumor cells. Despite remarkable results in some patients, ICIs do not demonstrate the same efficacy across all individuals. In this study, we present the first side-by-side comparison of an agent-based model (ABM) with an ordinary differential equation (ODE) model for ICIs targeting the PD-1/PD-L1 immune checkpoint. We consider tumor cells of high and low antigenicity and two distinct immune-cell kill mechanisms. Using key parameters calibrated from mouse bladder cancer studies, we simulate virtual tumors using both models. Our research identifies crucial tumor-immune characteristics that influence the efficacy of ICIs. By exploring the unique spatial insights provided by the ABM, we underscore the importance of considering the spatial complexity of the tumor microenvironment in mathematical models of ICIs, potentially paving the way for more effective cancer treatments. Since the introduction of the first immune checkpoint inhibitor (ICI), immunotherapy has changed the landscape of molecular therapeutics for cancers. However, ICIs do not work equally well on all cancers and for all patients. There has been a growing interest in using mathematical and computational models to optimize clinical responses. Ordinary differential equations (ODEs) have been widely used for mechanistic modeling in immuno-oncology and immunotherapy. They allow rapid simulations of temporal changes in the cellular and molecular populations involved. Nonetheless, ODEs cannot describe the spatial structure in the tumor microenvironment or quantify the influence of spatially-dependent characteristics of tumor-immune dynamics. For these reasons, agent-based models (ABMs) have gained popularity because they can model more detailed phenotypic and spatial heterogeneity that better reflect the complexity seen in vivo. In the context of anti-PD-1 ICIs, we compare treatment outcomes simulated from an ODE model and an ABM to show the importance of including spatial components in computational models of cancer immunotherapy. We consider tumor cells of high and low antigenicity and two distinct cytotoxic T lymphocyte (CTL) killing mechanisms. The preferred mechanism differs based on the antigenicity of tumor cells. Our ABM reveals varied phenotypic shifts within the tumor and spatial organization of tumor and CTLs despite similarities in key immune parameters, initial simulation conditions, and early temporal trajectories of the cell populations. [ABSTRACT FROM AUTHOR]

We develop and substantiate the algorithm for finding an ordinary differential equation of minimum order with polynomial coefficients over the field of rational numbers, whose solutions are a given system of polynomials (here, a system of the modified Laguerre–Cayley polynomials). [ABSTRACT FROM AUTHOR]

This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model’s parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term. [ABSTRACT FROM AUTHOR]

Alzheimer’s disease is one of the most prevalent types of dementia worldwide. It is caused by the accumulation of amyloid-beta (Aβ) plaques in the brain, disrupting communication pathways and memory. Microglia and astrocytes act as the immune system of the brain, clearing Aβ plaque deposits. However, these cells can lose effectiveness when Aβ plaque accumulation exceeds normal limits, leading to inflammation induced by proinflammatory cytokines. One type of treatment involves anti-Aβ drug therapy. Anti-Aβ drugs are believed to have the ability to reduce Aβ plaque deposits effectively. The mechanism of Aβ plaque accumulation can be explained by ordinary differential equations describing the growth of Aβ monomers. In this study, we aimed to develop a new mathematical model to elucidate the role of the immune system and drug therapy in reducing Aβ plaque deposits. Based on the simulation results, we conclude that the use of anti-Aβ drug therapy can decrease the concentration of Aβ plaque deposits, and the effective treatment duration for Alzheimer’s patients is estimated to be approximately 4 months starting from the time the drug was first administered.

This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model's parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term.

In this paper, we introduce second derivative multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and col-location methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order deriva-tives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems. [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]

This paper aims to identify exchange coefficients of a nonlinear polynomial tri-compartmental catenary system of (α + β) order. This is based on two principal procedures. The first procedure presented is related to the recommended solution consisting of introducing an adequate time t* > 0 in a way to be defined. That is to say: wait a moment to allow the exchange to settle in the polynomial (α + β) order nonlinear catenary system after injecting the quantity into the main compartment, then measure this compartment with compartment 2, at this time t* > 0. In the second procedure, we apply the Taylor formula to linearize the nonlinear system and identify the exchange coefficients. In the end, we will prove that the linearization method is stable. [ABSTRACT FROM AUTHOR]

In this article, the Upadhyaya transform is employed in diverse chemical reaction models expressed through ordinary differential equations. The investigation reveals that this transform provides precise and efficient solutions, circumventing the necessity for complex computations. Furthermore, the integration of graphical representations enhanced the interpretability of results, offering visual insights into the temporal evolution of reactant concentrations. These findings collectively underscore the efficacy of the Upadhyaya transform in addressing ordinary differential equations within chemical reaction models. [ABSTRACT FROM AUTHOR]

Physical phenomena are typically described in different areas of knowledge using mathematical models. These models permit the assessment of variables associated with the process without building experimental setups. In order to achieve this goal, it is necessary to formulate a differential equation that, depending on the number of variables, can be partial or ordinary type. In this context, the present work evaluates the mass balance for an open system from a numerical and mathematical perspective, and then experimentally validates the results obtained. The analytical solutions of the mathematical models are performed through variable separation, while the numerical solutions are obtained from the fourth-order Runge-Kutta method. The experimental results confirm that the applied differential equation accurately represents the process under consideration and predicts the filling times at which the system reaches the steady-state condition. Furthermore, it describes the system's transient process with an error of less than 4%. Moreover, the influence of operational variables, including the diameters and thermophysical properties of the fluid, on the time required for the system to reach a steady state is quantified numerically. Finally, the present study has proved that how an actual situation reproduced experimentally can be modeled using differential equations and how this strategy facilitates the learning and applying theoretical models in mathematical learning environments. [ABSTRACT FROM AUTHOR]

The present work is devoted to construction of the optimal interpolation formula exact for trigonometric functions sin(ωx) and cos(ωx). Here the analytical representations of the coefficients of the optimal interpolation formula in a certain Hilbert space are obtained using the discrete analogue of the differential operator. Taking the coefficients of the optimal interpolation formula as basis functions, in the finite element methods the boundary value problems for ordinary differential equations of the second order are approximately solved. In particular, it is shown that the coefficients of the optimal interpolation formula can serve as a set of effective basis functions. Approximate solutions of the differential equations are compared using the constructed basis functions and known basis functions. In particular, we have obtained numerical results for the cases when the numbers of basis functions are 6 and 11. In both cases, we have got that the accuracy of the approximate solution to the boundary value problems for second-order ordinary differential equations found using our basis functions is higher than the accuracy of the approximate solution found using known basis functions. It is proven that the accuracy of the approximate solution increases with increasing the number of basis functions.

Deep Feedforward Neural Networks (FNNs) with skip connections have revolutionized various image recognition tasks. In this paper, we propose a novel architecture called bidirectional FNN (BiFNN), which utilizes skip connections to aggregate features between its forward and backward paths. The BiFNN accepts any FNN as a plugin that can incorporate any general FNN model into its forward path, introducing only a few additional parameters in the cross-path connections. The backward path is implemented as a nonparameter layer, utilizing a discretized form of the neural memory Ordinary Differential Equation (nmODE), which is named ϵ -net. We provide a proof of convergence for the ϵ -net and evaluate its initial value problem. Our proposed architecture is evaluated on diverse image recognition datasets, including Fashion-MNIST, SVHN, CIFAR-10, CIFAR-100, and Tiny-ImageNet. The results demonstrate that BiFNNs offer significant improvements compared to embedded models such as ConvMixer, ResNet, ResNeXt, and Vision Transformer. Furthermore, BiFNNs can be fine-tuned to achieve comparable performance with embedded models on Tiny-ImageNet and ImageNet-1K datasets by loading the same pretrained parameters. [ABSTRACT FROM AUTHOR]

In this paper, diagonally implicit two–step peer methods for the numerical solution of initial value problems of order ordinary differential are divided into four types including the combination of explicit and implicit methods in a sequential or parallel environments. In this class of the methods, construction of implicit methods equipped with Runge–Kutta stability property together with A – or L –stability are investigated and examples of such methods are given up to order five. Finally, the efficiency and accuracy of the proposed methods are verified by applying them on some well–known stiff problems. [ABSTRACT FROM AUTHOR]

In this article, we investigated new travelling wave solutions for the modified Burgers equation with additional time-dependent variable coefficient via the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions. All solutions of this equation have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. The exact solutions have its great importance to reveal the internal mechanism of the physical phenomena. This method presents a wider applicability for handling nonlinear wave equations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce second derivative multistep collocation methods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and collocation methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order derivatives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the collocation methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expectations. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods. [ABSTRACT FROM AUTHOR]

Nonlinear projection equations (NPEs) provide a unified framework for solving various constrained nonlinear optimization and engineering problems. This paper presents a deep learning approach for solving NPEs by incorporating neurodynamic optimization and physics‐informed neural networks (PINNs). First, we model the NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization, and the objective becomes solving this ODE system. Second, we use a modified PINN to serve as the solution for the ODE system. Third, the neural network is trained using a dedicated algorithm to optimize both the ODE system and the NPE. Unlike conventional numerical integration methods, the proposed approach predicts the end state without computing all the intermediate states, resulting in a more efficient solution. The effectiveness of the proposed framework is demonstrated on a collection of classical problems, such as variational inequalities and complementarity problems. [ABSTRACT FROM AUTHOR]

The homotopy perturbation method (HPM) is one of the recent fundamental methods for solving differential equations. However, checking the accuracy of this method has been ignored by some authors in the literature. This paper reanalyzes the nonlinear system of ordinary differential equations (ODEs) describing the SIR epidemic model, which has been solved in the literature utilizing the HPM. The main objective of this work is to obtain a highly accurate analytical solution for this model via a direct technique. The proposed technique is mainly based on reducing the given system to a single nonlinear ODE that can be easily solved. Numerical results are conducted to compare our approach with the previous HPM, where the Runge–Kutta numerical method is chosen as a reference solution. The obtained results reveal that the current technique exhibits better accuracy over HPM in the literature. Moreover, some physical properties are introduced and discussed in detail regarding the influence of the transmission rate on the behavior of the SIR model. [ABSTRACT FROM AUTHOR]

In this article, the effects of physical flow parameters on squeezed fluid between parallel plates are explored through the Darcy porous channel when fluid is moving as a result of the upper plate being squeezed towards the stretchable lower plate, such as velocity slip, thermal slip, solutal slip, thermal stratification parameter, solutal stratification parameter, squeezing number, Darcy number, Prandtl number, and Schmidt number. The governing equations are transformed into a nonlinear ordinary differential equation using the appropriate similarity transformations. The resulting equations are solved by using the perturbation iteration method (PIT) to produce a convergent analytical solution with high accuracy. The phenomena of the squeezing fluid as the plates are moving apart and when they are coming together are illustrated using the resulting analytical solutions. Plots are used to discuss the significant effects of physical parameters on velocity, temperature, and fluid concentration profiles. The skin friction coefficient and Nusselt Sherwood values have graphical interpretations that are listed. For strong velocity slip parameters, the results demonstrate the existence of a minimum velocity profile close to the plate and a growing velocity profile distant from the plate. Additionally, as the slip effects rise, the fluid temperature and concentration both considerably drop. The results of the fourth-order Runge-Kutta method (RK4M) and the presented analytical solutions provided are in excellent agreement. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear dynamical system of autonomous ordinary differential equations with a small parameter such that two variables and are fast and another one is slow. If we take the limit as , then this becomes a "degenerate system" included in the one-parameter family of two-dimensional subsystems of fast motions with the parameter in some interval. It is assumed that in each subsystem there exists a structurally stable limit cycle . In addition, in the complete dynamical system there is some structurally stable periodic orbit that tends to a limit cycle for some as tends to zero. We can define the first return map, or the Poincaré map, on a local cross section in the hyperplane orthogonal to at some point. We prove that the Poincaré map has an invariant manifold for the fixed point corresponding to the periodic orbit on a guaranteed interval over the variable , and the interval length is separated from zero as tends to zero. The proved theorem allows one to formulate some sufficient conditions for the existence and/or absence of multipeak oscillations in the complete dynamical system. As an example of application of the obtained results, we consider some kinetic model of the catalytic reaction of hydrogen oxidation on nickel. [ABSTRACT FROM AUTHOR]

Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code. [ABSTRACT FROM AUTHOR]

The spectral collocation method for solving two-point boundary value problems for second order differential equations is implemented, based on representing the solution as an expansion in Chebyshev polynomials. The approach allows a stable calculation of both the spectral representation of the solution and its pointwise representation on any required grid in the definition domain of the equation and additional conditions of the multipoint problem. For the effective construction of SLAE, the solution of which gives the desired coefficients, the Chebyshev matrices of spectral integration are actively used. The proposed algorithms have a high accuracy for moderate-dimension systems of linear algebraic equations. The matrix of the system remains well-conditioned and, with an increase in the number of collocation points, allows finding solutions with ever-increasing accuracy.

Malaria importation is one of the hypothetical drivers of malaria transmission dynamics across the globe. Several studies on malaria importation focused on the effect of the use of conventional malaria control strategies as approved by the World Health Organization (WHO) on malaria transmission dynamics but did not capture the effect of the use of traditional malaria control strategies by vigilant humans. In order to handle the aforementioned situation, a novel system of Ordinary Differential Equations (ODEs) was developed comprising the human and the malaria vector compartments. Analysis of the system was carried out to assess its quantitative properties. The novel computational algorithm used to solve the developed system of ODEs was implemented and benchmarked with the existing Runge-Kutta numerical solution method. Furthermore, simulations of different vigilant conditions useful to control malaria were carried out. The novel system of malaria models was well-posed and epidemiologically meaningful based on its quantitative properties. The novel algorithm performed relatively better in terms of model simulation accuracy than Runge-Kutta. At the best model-fit condition of 98% vigilance to the use of conventional and traditional malaria control strategies, this study revealed that malaria importation has a persistent impact on malaria transmission dynamics. In lieu of this, this study opined that total vigilance to the use of the WHO-approved and traditional malaria management tools would be the most effective control strategy against malaria importation.

The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters. [ABSTRACT FROM AUTHOR]

Despite the great promise of machine learning in the structural seismic analysis, the deployment of advanced neural networks has been limited in practical applications because of the high costs of data acquisition. This paper introduces a new framework that integrates the powerful learning ability of physics-informed neural networks (PINNs) with the effectiveness of pseudo-labelling in data augmentation to improve the accuracies of seismic response predictions of structures. The architecture of PINNs consists of two blocks of gated recurrent unit-fully connected neural networks (GRU-FCNNs) and one block of ordinary differential equation (ODE) that leverages the knowledge of structural dynamics. The first block of GRU-FCNNs serves as a generator of pseudo-labels when drawing upon the input of unlabelled datasets. The second block of GRU-FCNNs in combination with the ODE block is a selector of reliable pseudo-labels. The performance of PINNs trained on limited labelled data can be significantly improved by successively selecting reliable pseudo-labels from the generator and selector to supplement training datasets. The effectiveness of pseudo-labelling in PINNs is validated and compared with PINNs without pseudo-labelling through case studies with simulation datasets and real datasets from experiment. The results show that the proposed framework is effective and robust in improving prediction accuracies of structural seismic responses with limited labelled data. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov–Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)- C ∞ -smooth and (non)-analytic periodic center manifolds. [ABSTRACT FROM AUTHOR]

Malaria importation is one of the hypothetical drivers of malaria transmission dynamics across the globe. Several studies on malaria importation focused on the effect of the use of conventional malaria control strategies as approved by the World Health Organization (WHO) on malaria transmission dynamics but did not capture the effect of the use of traditional malaria control strategies by vigilant humans. In order to handle the aforementioned situation, a novel system of Ordinary Differential Equations (ODEs) was developed comprising the human and the malaria vector compartments. Analysis of the system was carried out to assess its quantitative properties. The novel computational algorithm used to solve the developed system of ODEs was implemented and benchmarked with the existing Runge-Kutta numerical solution method. Furthermore, simulations of different vigilant conditions useful to control malaria were carried out. The novel system of malaria models was well-posed and epidemiologically meaningful based on its quantitative properties. The novel algorithm performed relatively better in terms of model simulation accuracy than Runge-Kutta. At the best model-fit condition of 98% vigilance to the use of conventional and traditional malaria control strategies, this study revealed that malaria importation has a persistent impact on malaria transmission dynamics. In lieu of this, this study opined that total vigilance to the use of the WHO-approved and traditional malaria management tools would be the most effective control strategy against malaria importation. [ABSTRACT FROM AUTHOR]

This article introduces an Ordinary Differential Equation (ODE) notion for survival analysis. The ODE notion not only provides a unified modeling framework, but more importantly, also enables the development of a widely applicable, scalable, and easy-to-implement procedure for estimation and inference. Specifically, the ODE modeling framework unifies many existing survival models, such as the proportional hazards model, the linear transformation model, the accelerated failure time model, and the time-varying coefficient model as special cases. The generality of the proposed framework serves as the foundation of a widely applicable estimation procedure. As an illustrative example, we develop a sieve maximum likelihood estimator for a general semiparametric class of ODE models. In comparison to existing estimation methods, the proposed procedure has advantages in terms of computational scalability and numerical stability. Moreover, to address unique theoretical challenges induced by the ODE notion, we establish a new general sieve M-theorem for bundled parameters and show that the proposed sieve estimator is consistent and asymptotically normal, and achieves the semiparametric efficiency bound. The finite sample performance of the proposed estimator is examined in simulation studies and a real-world data example. for this article are available online. [ABSTRACT FROM AUTHOR]

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems.

In view of the increasing system management requirements of garden landscapes in the construction of smart cities, this paper combines ordinary differential equations with big data and other information technologies. Design ideas of smart city garden big data information management and display system under differential equation. By analyzing the research literature, combined with space syntax, network text analysis and other methods, the theoretical framework of urban landscape planning method based on ordinary differential equations based on big data is constructed, and the aspect planning of urban garden landscape information management module, three-dimensional display module and recommendation module is determined. content. The experimental results show that the user interaction rate when using the system of this paper for information management of urban gardens shows a significant upward trend compared with the traditional system, and the average user interaction rate is higher than 40%. The application of ordinary differential equations and big data technology guides participatory planning and promotes the improvement of the information management level of smart city garden landscape. The designed information processing system has higher user experience and comfort.

A basic mathematical model in epidemiology is the SIR (Susceptible–Infected–Removed) model, which is commonly used to characterize and study the dynamics of the spread of some infectious diseases. In humans, the time scale of a disease can be short and not necessarily fatal, but in some animals (for example, insects) this same short time scale can make the disease fatal if we take into account their life expectancy. In this work, we will see how a positive feedback effect (decrease of the susceptible population at small densities) in a SIR model can cause a qualitative characterization of the dynamics defined by the original SIR model. Finally, we will also show with numerical simulations how a delay in the feedback effect causes very interesting qualitative changes of the system with epidemiological significance.

The objective of the research is to estimate the transmission rate of an infection (β) in the SI epidemical model, using Bayesian statistical methods from observed data in Peru. After studying the SI mathematical model and Bayesian statistical inference metho'ds, a Bayesian estimator is proposed to estimate the transmisión rate of an infection in this model and a procedure is proposed to estimate this rate using Montecarlo simulation based on Markov chains - MCMC.

Blood-brain barrier (BBB) permeability prediction plays a pivotal role in drug discovery for neurological disorders which is essential for the development of central nervous system (CNS) drugs. Compounds having high permeability must be found to synthesize brain drugs for the treatment of different brain disorders such as Alzheimer's, Parkinson's, and brain tumors. Developing an accurate mathematical computational model to determine the exact brain permeability value for a possible drug is essential for advancing and improving the success rate of the development of drugs for neurological treatment. We developed a combined method capable of forecasting the logBB value of the compound in question for lightBBB and B3DB datasets by using the Convolutional neural network (CNNs), Recurrent Neural Networks (RNNs) and Ordinary Differential Equations (ODEs). The results demonstrate the overall assessment of the prediction ability of BBB permeability. CNN-LSTM model tests the performance for the prediction of logBB values in two different datasets LightBBB and B3DB. In both datasets, CNN-LSTM achieves lower RMSE values as compared to other models, showing that it has better predictive performance. Specifically, in the LightBBB dataset, the CNN-LSTM model achieves an RMSE of 0.59, while in the B3DB datasets, it achieves an even lower RMSE of 0.85. CNN-LSTM model shows highly effective in accurately predicting logBB values in both datasets.

This research aims to propose a numerical methodology that employs a non-increasing sequence to approximate the solution for a large class of ordinary differential equations like the ones described in complex non-linear heat transfer through porous fins. It also aims to compare this proposed methodology with an earlier one using a non-decreasing sequence of elements and to provide an upper-bound estimation for the error. The comparison between the non-increasing and non-decreasing sequences showed excellent agreement when applied to an example of convection and radiation in porous fins.

This paper introduces a novel neural network approach based on Lie groups to effectively solve initial value problems of differential equations for nonlinear dynamical systems. Our method utilizes a priori knowledge inherent in the system, i.e. Lie group expressions, and employs a single-layer network structure with the essence of a multilayer perceptron (MLP). To validate the effectiveness of our approach, we conducted an extensive empirical study using various examples representing complex nonlinear dynamical systems. The research results demonstrate the outstanding performance and efficacy of our method, outperforming Neural Ordinary Differential Equations in terms of accuracy, convergence speed, and stability. [ABSTRACT FROM AUTHOR]

This paper proposes a deep learning approach for solving non-smooth convex optimization problems (NCOPs), which have broad applications in computer science, engineering, and physics. Our approach combines neurodynamic optimization with physics-informed neural networks (PINNs) to provide an efficient and accurate solution. We first use neurodynamic optimization to formulate an initial value problem (IVP) that involves a system of ordinary differential equations for the NCOP. We then introduce a modified PINN as an approximate state solution to the IVP. Finally, we develop a dedicated algorithm to train the model to solve the IVP and minimize the NCOP objective simultaneously. Unlike existing numerical integration methods, a key advantage of our approach is that it does not require the computation of a series of intermediate states to produce a prediction of the NCOP. Our experimental results show that this computational feature results in fewer iterations being required to produce more accurate prediction solutions. Furthermore, our approach is effective in finding feasible solutions that satisfy the NCOP constraint. [ABSTRACT FROM AUTHOR]

The effective sufficient conditions are established for the numerical approximation of solutions of the initial problem with weight for linear systems of ordinary differential equations with singularities. [ABSTRACT FROM AUTHOR]