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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]

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]

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]

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 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]

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 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]

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]

In this paper, we present a deterministic mathematical model for the study of overweight, and obesity in a population and its impact on the growth of the number of diabetics. For the construction of the model, we take into account social factors and the interactions between the elements of society. We find the basic reproduction number and prove the global stability of the disease-free equilibrium point. We present theoretical results and find the sensitivity indices to characterize the impact of parameters associated with overweight, obesity and diagnosed diabetes on the basic reproduction number. To validate the model, we perform computational simulations and study the basic reproduction number and compartments. We present the behavior of the compartments for a scenario and study the impact of the variation of parameters associated with overweight by social pressure and diabetes due to causes other than obesity. [ABSTRACT FROM AUTHOR]

This paper investigates the possibility of automatically linearizing nonlinear models. Constructing a linearised model for a nonlinear system is quite labor-intensive and practically unrealistic when the dimension is greater than 3. Therefore, it is important to automate the process of linearisation of the original nonlinear model. Based on the application of computer algebra, a constructive algorithm for the linearisation of a system of non-linear ordinary differential equations was developed. A software was developed on MatLab. The effectiveness of the proposed algorithm has been demonstrated on applied problems: an unmanned aerial vehicle dynamics model and a twolink robot model. The obtained linearized models were then used to test the stability of the original models. In order to account for possible inaccuracies in the measurements of the technical parameters of the model, an interval linearized model is adopted. For such a model, the procedure for constructing the corresponding interval characteristic polynomial and the corresponding Hurwitz matrix is automated. On the basis of the analysis of the properties of the main minors of the Hurwitz matrix, the stability of the studied system was analyzed. [ABSTRACT FROM AUTHOR]

The Information Bottleneck (IB) is a method of lossy compression of relevant information. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information embedded in the input. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory when input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB's intimate relations with RD, we provide substantial insights into its solution structure, highlighting caveats in its finite-dimensional treatments. Sub-optimal solutions are seen to collide or exchange optimality at its bifurcations. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB's first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, we not only detect IB bifurcations but also identify their type in order to handle them accordingly. Rather than approaching the IB's optimal tradeoff curve from sub-optimal directions, the latter allows us to follow a solution's trajectory along the optimal curve under mild assumptions. We thereby translate an understanding of IB bifurcations into a surprisingly accurate numerical algorithm. [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.

In this paper, we present a deterministic mathematical model for the study of overweight, and obesity in a population and its impact on the growth of the number of diabetics. For the construction of the model, we take into account social factors and the interactions between the elements of society. We find the basic reproduction number and prove the global stability of the disease-free equilibrium point. We present theoretical results and find the sensitivity indices to characterize the impact of parameters associated with overweight, obesity and diagnosed diabetes on the basic reproduction number. To validate the model, we perform computational simulations and study the basic reproduction number and compartments. We present the behavior of the compartments for a scenario and study the impact of the variation of parameters associated with overweight by social pressure and diabetes due to causes other than obesity.

In this article, we investigate the precise expression forms of entire solutions for two certain Fermat-type ordinary differential equations: (a0f+a1f′)2+(a0f+a2f′)2=p{\left({a}_{0}f+{a}_{1}{f}^{^{\prime} })}^{2}+{\left({a}_{0}f+{a}_{2}{f}^{^{\prime} })}^{2}=p and (a0f+a1f′)2+(a0f+a2f′)2=eg\hspace{0.39em}{\left({a}_{0}f+{a}_{1}{f}^{^{\prime} })}^{2}+{\left({a}_{0}f+{a}_{2}{f}^{^{\prime} })}^{2}={e}^{g} in C{\mathbb{C}} by making use of the Nevanlinna theory for meromorphic functions, where a0{a}_{0}, a1{a}_{1}, and a2{a}_{2} are the complex numbers with a0≠0{a}_{0}\ne 0 and ∣a1∣+∣a2∣≠0| {a}_{1}| +| {a}_{2}| \ne 0, while pp and gg are the polynomials in C{\mathbb{C}}. Moreover, some examples are given to illustrate the existence of entire solutions for the aforementioned two certain equations.

It is known that the Dhage iteration method is very much useful for proving the existence and approximation results for nonlinear hybrid differential and integral equations. In this paper, we introduce a notion of the local solution for the initial value problems of nonlinear second order ordinary differential equations and establish a couple of approximation results for local existence and uniqueness of the solution via Dhage monotone iteration method. Again, various hybrid fixed point theorems are involved in the Dhage iteration method as per the demand of the nonlinear hybrid equations. Here, we base our Dhage monotone iteration method on the recent hybrid fixed point theorems of Dhage (2022) and Dhage et al. (2022). There are different notions of stability of the nonlinear equations. Here we discuss the Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established by construction of an algorithm. Finally, our main abstract results are also illustrated with a couple of numerical examples. [ABSTRACT FROM AUTHOR]

Introduction: Dynamic crop growth models are an important tool to predict complex traits, like crop yield, for modern and future genotypes in their current and evolving environments, as those occurring under climate change. Phenotypic traits are the result of interactions between genetic, environmental, and management factors, and dynamic models are designed to generate the interactions producing phenotypic changes over the growing season. Crop phenotype data are becoming increasingly available at various levels of granularity, both spatially (landscape) and temporally (longitudinal, time-series) from proximal and remote sensing technologies. Methods: Here we propose four phenomenological process models of limited complexity based on differential equations for a coarse description of focal crop traits and environmental conditions during the growing season. Each of these models defines interactions between environmental drivers and crop growth (logistic growth, with implicit growth restriction, or explicit restriction by irradiance, temperature, or water availability) as a minimal set of constraints without resorting to strongly mechanistic interpretations of the parameters. Differences between individual genotypes are conceptualized as differences in crop growth parameter values. Results: We demonstrate the utility of such low-complexity models with few parameters by fitting them to longitudinal datasets from the simulation platform APSIM-Wheat involving in silico biomass development of 199 genotypes and data of environmental variables over the course of the growing season at four Australian locations over 31 years. While each of the four models fits well to particular combinations of genotype and trial, none of them provides the best fit across the full set of genotypes by trials because different environmental drivers will limit crop growth in different trials and genotypes in any specific trial will not necessarily experience the same environmental limitation. Discussion: A combination of low-complexity phenomenological models covering a small set of major limiting environmental factors may be a useful forecasting tool for crop growth under genotypic and environmental variation. [ABSTRACT FROM AUTHOR]

This paper re-analyzes the nonlinear Susceptible–Infected–Recovered (SIR) model using a hybrid approach based on the Laplace–Padé technique. The proposed approach is successfully applied to extract several analytic approximations for the infected and recovered individuals. The domains of applicability of such analytic approximations are addressed. In addition, the present results are validated through various comparisons with the Runge–Kutta numerical method. The obtained analytical results agree with the numerical ones for a wide range of numbers of contacts featured in the studied model. The efficiency of the present analysis reveals that it can be implemented to deal with other systems describing real-life phenomena. [ABSTRACT FROM AUTHOR]

This study deals with the numerical solution of a class of linear systems of second-order boundary value problems (BVPs) using a new symmetric cubic B-spline method (NCBM). This is a typical cubic B-spline collocation method powered by new approximations for second-order derivatives. The flexibility and high order precision of B-spline functions allow them to approximate the answers. These functions have a symmetrical property. The new second-order approximation plays an important role in producing more accurate results up to a fifth-order accuracy. To verify the proposed method's accuracy, it is tested on three linear systems of ordinary differential equations with multiple step sizes. The numerical findings by the present method are quite similar to the exact solutions available in the literature. We discovered that when the step size decreased, the computational errors decreased, resulting in better precision. In addition, details of maximum errors are investigated. Moreover, simple implementation and straightforward computations are the main advantages of the offered method. This method yields improved results, even if it does not require using free parameters. Thus, it can be concluded that the offered scheme is reliable and efficient. [ABSTRACT FROM AUTHOR]

This article applies the variational Kantorovich-Vlasov method to obtain exact mathematical solutions to the bending problem of uniformly loaded thin plate with two opposite simply supported edges and two free edges. The studied problem is a common theme in the analysis and design of structures: Vlasov method was adopted simultaneously in the variational Kantorovich method, and the deflection function w(x, y) is expressed in variable-separable form as single infinite series in terms of the unknown function g(y) and known sinusoidal functions of x coordinate variable f(x) where f(x) satisfies Dirichlet boundary conditions at the simple supports. The total potential energy functional , expressed in terms of g(y) and the derivatives g(y), g(y) is then minimized with respect to g(y) using the Euler-Lagrange differential equations. The resulting equation of equilibrium is a set of fourth order inhomogeneous ordinary differential equations (ODEs) in g(y). The general solution is found as a single series of infinite terms and boundary conditions are enforced to find the integration constants. The single infinite series expression found for w(x, y) satisfies the governing equations at every point on the domain and the boundaries and is thus exact within the scope of thin plate theory adopted to idealize the plate. Moment-deflection equations are used to obtain exact analytical expressions for the bending moments Mxx, Myy. Deflection and bending moments are computed at the plate center; as well as at the middle of the free edges. Comparison of the plate center deflections and bending moments for various aspect ratios illustrate that the exact solutions by the present work are in agreement with Levy solutions presented by Timoshenko and Woinowsky-Krieger and symplectic elasticity solutions presented by Cui Shuang. The present results for bending moments at the free edges for various aspect ratios agree with the Levy results presented by Timoshenko and Woinowsky-Krieger and symplectic elasticity results presented by Cui Shuang. The novelty of the work is that to the author’s knowledge this is the first application of the variational Kantorovich-Vlasov method to the formulation of exact analytical solution to bending analysis of thin rectangular plates with SFrSFr boundaries. [ABSTRACT FROM AUTHOR]

The struggle to find the analytic solution of several differential equations leads to several issues like difficulty in finding solutions, singularities, convergent issues, and stability. Because of these problems, most of the researchers come up with explicit approaches such as the Runge Kutta method, Euler’s method, and Taylor’s polynomial method for finding numerical solutions to the ordinary differential equation. In this work, we combine both the Aitken methods and Newton’s interpolation method (NIM) to solve first-order differential equations. The numerical results obtained provide minimal error. The result is supported by solving an example.

In order to study the dynamic characteristics of the interest rate term structure of zero-yield bonds in China's bond market, and why China's interest rate term shows such changing characteristics 96-month interest rate structure data at the end of December. Among them, the term structure of interest rates in the securities market is from 1 year to 20 years, and the term structure of interest rates in the interbank market is from one year to 30 years. Analyze the data to find out the basic characteristics of the dynamic change of the term structure of China's interest rate, and find the entire term structure of China's bond market from the perspective of trends. This feature is evident in short-term interest rates in the interbank market, influenced by central bank financial policies. In terms of statistical characteristics, as the yield to maturity of Chinese bonds increases on average, that is, the inclination of the term structure of interest rates increases. The volatility of yield to maturity of short-term bonds is significantly higher than that of long-term bonds, and the volatility of yield to maturity in the securities market is lower than that of the interbank market, mainly because short-term bonds and the interbank market are easily affected by the central bank’s financial policies. Experiments show that the level of inflation rate (PPI growth rate) is an important factor that determines the level of the term structure of interest rates. Inflation rate and interest rate levels are important determinants of skew and curvature factors.

Based on the theory of ordinary differential equations, this paper proposes a stable and discriminative method for piano tone tuning. We perform discriminative training on the hidden Markov tone model according to ordinary differential equations’ feature extraction parameters and model parameters. The model can improve the recognition rate of piano tones. This paper trains and tests the MAPS universal dataset for musical transcription of automatic piano tones. The model is uniformly trained on the synthetic part and tested on the real recording part. The experimental study found that the proposed model has high recognition stability and accuracy in piano tone recognition.

In order to expand the basic theory of boundary value problems of fractional differential equations,the existence of solutions of double-order Hilfer fractional differential equations under Riemann-Stieltjes integral boundary conditions under resonance conditions was studied.Firstly,two suitable Banach spaces were constructed.Secondly,appropriate operators in Banach spaces were defined and Mawhin′s coincidence theory was used to prove the existence of the solution to the double-order Hilfer fractional resonance boundary value problem.Finally,an example was given to illustrate the correctness of the results.The results show that the solution of the double-order Hilfer fractional resonant boundary value problem exists in a suitable Banach space.Using Mawhin′s coincidence degree theory to study the existence of solutions to the double-order Hilfer fractional resonance boundary value problem expands the range of orders in the differential operator,enriches the theory of solvability of fractional differential equations and provides an important theoretical basis for the study of fractional differential equations in aerodynamics,economics,control theory and other fields.

This work shows the voltage regulation of a DC–DC buck converter by applying sliding mode control using three different cases of sliding surfaces. The DC–DC buck converter is modeled by ordinary differential equations (ODEs) that are solved by applying numerical methods. The ODEs describe two state variables that are associated to the capacitor voltage and the inductor current. The state variable associated to voltage is regulated by applying two well-known sliding surfaces and a third one that is introduced herein to improve the response of the sliding mode control. The stability of the proposed sliding surface is verified by using a Lyapunov theorem to guarantee closed-loop stability. Finally, simulation results show the improvement of voltage regulation when applying the proposed sliding surface compared to already reported approaches. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of mappings called "generalized β - ϕ -Geraghty contraction-type mappings". We use our new class to formulate and prove some coupled fixed points in the setting of partially ordered metric spaces. Our results generalize and unite several findings known in the literature. We also provide some examples to support and illustrate our theoretical results. Furthermore, we apply our results to discuss the existence and uniqueness of a solution to a coupled ordinary differential equation as an application of our finding. [ABSTRACT FROM AUTHOR]

IntroductionDynamic crop growth models are an important tool to predict complex traits, like crop yield, for modern and future genotypes in their current and evolving environments, as those occurring under climate change. Phenotypic traits are the result of interactions between genetic, environmental, and management factors, and dynamic models are designed to generate the interactions producing phenotypic changes over the growing season. Crop phenotype data are becoming increasingly available at various levels of granularity, both spatially (landscape) and temporally (longitudinal, time-series) from proximal and remote sensing technologies.MethodsHere we propose four phenomenological process models of limited complexity based on differential equations for a coarse description of focal crop traits and environmental conditions during the growing season. Each of these models defines interactions between environmental drivers and crop growth (logistic growth, with implicit growth restriction, or explicit restriction by irradiance, temperature, or water availability) as a minimal set of constraints without resorting to strongly mechanistic interpretations of the parameters. Differences between individual genotypes are conceptualized as differences in crop growth parameter values.ResultsWe demonstrate the utility of such low-complexity models with few parameters by fitting them to longitudinal datasets from the simulation platform APSIM-Wheat involving in silico biomass development of 199 genotypes and data of environmental variables over the course of the growing season at four Australian locations over 31 years. While each of the four models fits well to particular combinations of genotype and trial, none of them provides the best fit across the full set of genotypes by trials because different environmental drivers will limit crop growth in different trials and genotypes in any specific trial will not necessarily experience the same environmental limitation.DiscussionA combination of low-complexity phenomenological models covering a small set of major limiting environmental factors may be a useful forecasting tool for crop growth under genotypic and environmental variation.

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.

By combining the concept of orthogonality and the Geraghty type contraction, we give some fixed point results in the class of O-metric spaces. Our obtained results extend the existing results in the literature. We also resolve an ordinary type differential equation.

In this study, a numerical integrator that is based on a nonlinear interpolant, for the local representation of the theoretical solution is presented. The resulting integrator aims to solve second and higher-order initial value problems as systems of first-order initial value problems. The method is designed to have two complex functions as control parameters. The control parameters may become real, depending on the nature of the second-order initial value problems to be solved. The generalization and properties of the scheme are also presented. [ABSTRACT FROM AUTHOR]