Problems For Ordinary Differential Equations Research Articles

Intelligent Levenberg–Marquardt neural network solution to flow of carbon nanotubes in a nozzle of liquid rocket engine

In this research, we utilized artificial neural networks along with the Levenberg–Marquardt algorithm (ANN-LMA) to interpret numerical computations related to the efficiency of heat transfer in a regenerative cooling channel of a rocket engine. We used a mixture of Kerosene and carbon nanotubes (CNTs) for this purpose, examining both single-wall carbon nanotubes and multi-wall carbon nanotubes. The primary equations were converted into a dimensionless form using a similarity transformation technique. To establish a reference dataset for ANN- LMA and to analyze the movement and heat transfer properties of CNTs, we employed a numerical computation method called bvp4c, which is a solver for boundary value problems in ordinary differential equations using finite difference schemes combined with the Lobatto IIIA algorithm in MATLAB mathematical software. The ANN- LMA method was trained, tested and validated using these reference datasets to approximate the solutions of the flow model under different scenarios involving various significant physical parameters. We evaluated the accuracy of the proposed ANN- LMA model by comparing its results with the reference outcomes. We validated the performance of ANN- LMA in solving the Kerosene-based flow with CNTs in a rocket engine through regression analysis, histogram studies, and the calculation of the mean square error. The comprehensive examination of parameters undertaken in this research endeavor is poised to provide invaluable support to aerospace engineers as they endeavor to craft regenerative equipment with optimal efficiency. The pragmatic implications of our study are wide-ranging, encompassing domains as diverse as aerospace technology, materials science, and artificial intelligence. This research holds the potential to catalyze progress across multiple sectors and foster the evolution of increasingly efficient and sustainable systems.

A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE)

A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE)

PyPESTO: a modular and scalable tool for parameter estimation for dynamic models.

Mechanistic models are important tools to describe and understand biological processes. However, they typically rely on unknown parameters, the estimation of which can be challenging for large and complex systems. pyPESTO is a modular framework for systematic parameter estimation, with scalable algorithms for optimization and uncertainty quantification. While tailored to ordinary differential equation problems, pyPESTO is broadly applicable to black-box parameter estimation problems. Besides own implementations, it provides a unified interface to various popular simulation and inference methods. pyPESTO is implemented in Python, open-source under a 3-Clause BSD license. Code and documentation are available on GitHub (https://github.com/icb-dcm/pypesto).

On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of 4n order

The paper considers a two-point boundary value problem with homogeneous boundary conditions for a single nonlinear ordinary differential equation of order 4n. Using the well-known Krasnoselsky theorem on the expansion (compression) of a cone, sufficient conditions for the existence of a positive solution to the problem under consideration are obtained. To prove the uniqueness of a positive solution, the principle of compressed operators was invoked. In conclusion, an example is given that illustrates the fulfillment of the obtained sufficient conditions for the unique solvability of the problem under study.

Evolutionary Neural Networks for Option Pricing: Multi-Assets Option and Exotic Option

This paper presents a novel framework based on the evolutionary neural network to solve the generalized Black-Scholes equation arising in the financial market efficiently and accurately. We first employ evolutionary neural networks to parameterize the Partial Differential Equations (PDEs) involved in option pricing. This approach allows us to simplify the Black-Scholes PDEs and convert them into the corresponding Ordinary Differential Equations (ODEs). Thus we can use standard ODE solvers such as Euler’s method to solve the simplified ODE problem. The proposed framework is flexible and can handle various boundary conditions and terminal conditions, allowing customization based on specific market requirements and scenarios. Moreover, our method offers a reliable and deterministic solution methodology for the pricing framework, as it does not rely on stochastic training. Unlike other approaches that incorporate stochastic elements in their training process, our method eliminates the need for such training and provides consistent results. This deterministic nature enhances the reliability and stability of our approach, making it well-suited for real-world applica- tions in option pricing and financial markets. The experiments on multiple settings are carried out to illustrate the applicability and accuracy of the proposed framework.

Algorithmic realization of exact three‐point difference scheme for singular Sturm–Liouville problem

AbstractA new algorithmic realization of exact three‐point difference schemes for a class of singular Sturm–Liouville problems is presented. It is shown that to compute the coefficients of the exact scheme in an arbitrary grid node, it is necessary to solve two auxiliary initial value problems for the second order linear ordinary differential equations: one problem on the interval (forward) and one problem on the interval (backward). Finally, the coefficient stability of the exact three‐point difference scheme is proved.

In-context operator learning with data prompts for differential equation problems

This paper introduces the paradigm of "in-context operator learning" and the corresponding model "In-Context Operator Networks" to simultaneously learn operators from the prompted data and apply it to new questions during the inference stage, without any weight update. Existing methods are limited to using a neural network to approximate a specific equation solution or a specific operator, requiring retraining when switching to a new problem with different equations. By training a single neural network as an operator learner, rather than a solution/operator approximator, we can not only get rid of retraining (even fine-tuning) the neural network for new problems but also leverage the commonalities shared across operators so that only a few examples in the prompt are needed when learning a new operator. Our numerical results show the capability of a single neural network as a few-shot operator learner for a diversified type of differential equation problems, including forward and inverse problems of ordinary differential equations, partial differential equations, and mean-field control problems, and also show that it can generalize its learning capability to operators beyond the training distribution.

Nonlinear Singularly Perturbed Cauchy Problem With Inner Layer

In the article, the asymptotics of the solution of the Cauchy problem for a nonlinear autonomous ordinary differential equation of the first order, which is a simple mathematical model with an inner layer, is constructed. For all theoretical calculations, evidence-based calculations in the Maple system are given.

Numerical integration of the Cauchy problem with non-singular special points

Solutions of many applied Cauchy problems for ordinary differential equations have one or more multiple zeros on the integration segment. Examples are the equations of special functions of mathematical physics. The presence of multiples of zeros significantly complicates the numerical calculation, since such problems are ill-conditioned. Round-off errors may corrupt all decimal digits of the solution. Therefore, multiple zeros should be treated as special points of the differential equations. In the present paper, a local solution transformation is proposed, which converts the multiple zero into a simple one. The calculation of the latter is not difficult. This makes it possible to dramatically improve the accuracy and reliability of the calculation. Illustrative examples have been carried out, which confirm the advantages of the proposed method.

Useful Ideas on the Numerical Techniques Used for the Solution of the Two-Point Boundary Value Problems of Ordinary Differential Equations

Useful Ideas on the Numerical Techniques Used for the Solution of the Two-Point Boundary Value Problems of Ordinary Differential Equations

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem for a 4nth-Order Nonlinear Ordinary Differential Equation

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem for a 4nth-Order Nonlinear Ordinary Differential Equation

Steady flow of thin film over porous moving and non‐flat sheet with nonlinear kinematics of exponential type

AbstractA generalized model of flow of viscous thin film has been presented and the film is maintained over a porous, moving and non‐flat sheet. We categorically emphasized on the nonuniform and nonlinear kinematics of the sheet and deformation of thin film and variation of all quantities specified at the boundaries are taken of exponential type. The combined effects of deformation of both thin film and sheet along with the nonlinear kinematics of sheet have been analyzed on the characteristics of flow. The governing partial differential equations are transformed into ordinary differential equations (ODEs) by using similarity transformations and the final problem of ODEs is solved with the help of bvp4c technique, whereas, the result for the velocity and skin friction are graphed for different values of the injection (suction), stretching (shrinking) and deformation (contraction/expansion) of both thin film and sheet parameters. Note that the increasing, decreasing, uniform, linear, nonlinear and boundary layer behaviors of the velocity profiles and skin friction are noted for multiple choices of the parameters. Moreover, flows in upstream and downstream directions have been observed for different values and diverse nature of the parameters.

MAPLE SIMULATION CODES FOR STABILITY ANALYSIS OF VARIABLE STEP SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR INTEGRATING A SYSTEM OF FIRST ORDER STIFF IVPS

Strength of numerical scheme is rated by the properties it possessed and in turn the kind of problems it can handle. Zero stable method can effectively handle ODEs problem. While, an A – stable method can solve stiff ODEs problem. Analyzing stability of block methods are been carried out using various software. This work aimed at using simplified Maple simulation code to critically analyze avariable step size multi-block backward differentiation formula for the solution stiff initial value problems of ordinary differential equations. The Graphical comparisons of the simulated result obtained is made using Matlab to depict the performing schemes.

Exploiting the Abstract Calculus Pattern for the Integration of Ordinary Differential Equations for Dynamics Systems: An Object-Oriented Programming Approach in Modern Fortran

This manuscript relates to the exploiting of the abstract calculus pattern (ACP) for the (numerical) solution of ordinary differential equation (ODEs) systems, which are ubiquitous mathematical formulations of many physical (dynamical) phenomena. We present FOODIE, a software suite aimed to numerically solve ODE problems by means of a clear, concise, and efficient abstract interface. The results presented prove manifold findings, in particular that our ACP approach enables ease of code development, clearness and robustness, maximization of code re-usability, and conciseness comparable with computer algebra system (CAS) programming (interpreted) but with the computational performance of compiled programming. The proposed programming model is also proven to be agnostic with respect to the parallel paradigm of the computational architecture: the results show that FOODIE applications have good speedup with both shared (OpenMP) and distributed (MPI, CAF) memory architectures. The present paper is the first announcement of the FOODIE project: the current implementation is extensively discussed, and its capabilities are proved by means of tests and examples.

On the Stability of a High Order Stiffly Stable Parameter Dependent Nested Hybrid Linear Multistep Methods for Stiff ODEs

This paper is on the stability of a high order stiffly stable parameter dependent nested hybrid multistep method for the numerical integration of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The method incorporates one or more off-step points for better stability properties. The stability properties of the methods were investigated and the intervals of absolute stability of the methods with step number $k \leq 6$ are presented using the boundary locus techniques. The method is $A$-stable and $A(\alpha)$-stable which makes the methods more suitable for stiff initial value problems.

Derivation of Five-Stage Implicit Runge-Kutta Method of Order 10 via Gauss-Legendre Quadrature for Ordinary Differential Equations

This paper presents developing and implementing a five-stage implicit Runge-Kutta method of order ten via Gauss-Legendre quadrature for stiff or oscillatory first-order initial value problems (IVPs) of ordinary differential equations (ODEs). Using continuous collocation and interpolation techniques, the implicit Runge-Kutta method was developed by combining Legendre polynomials and exponential functions as the basis function. The properties of the method were investigated, and it was shown that it is consistent and A-stable. The new method was evaluated on two sampled problems involving stiffness and oscillation. The numerical results demonstrate that the new implicit Runge-Kutta is computationally efficient and outperforms previous methods of similar derivations.

Naimark Problem for a Fractional Ordinary Differential Equation

Naimark Problem for a Fractional Ordinary Differential Equation

On Solutions of a Boundary Value Problem for a Second-Order Differential Equation with a Parameter and Discontinuous Right-Hand Side

A boundary value problem for a second-order ordinary differential equation with a parameter and discontinuous right-hand side is considered. Theorems on the number of solutions to the problem are established. The resulting solutions are illustrated by plots. The process of numerically solving the problem is described.

Analysis of the framework for realizing core literacy in teaching physical education and health curriculum based on the ordinary differential error approximation method

Abstract The key to implementing the fundamental task of establishing moral education is how the abstract and broad physical education and health curriculum core literacy is translated into concrete and visible physical education and health subject curriculum and teaching practice. This paper firstly explains the framework for the realization of core literacy in PE and health curriculum in teaching and learning, specifically from three perspectives: context, health behavior, and physical virtue. Secondly, the abstract core literacy of physical education and health curriculum is transformed into a side problem of ordinary differential equation with unknown parameter initial values by using the method of ordinary differential error approximation, which can obtain the parametric expressions of the approximation of function values at the nodes and realize the concretization. The experimental results show that there is a significant difference between the core literacy of physical education and health curriculum based on the ordinary differential error approximation method before and after the application of the core literacy in teaching, in which the comprehensive improvement of motor ability is 23.7%, the comprehensive improvement of health behavior is 19.2%, and the comprehensive improvement of physical morality is 26.6%. This paper successfully visualized the core literacy issues of physical education and health curriculum based on the ordinary differential error approximation method, which provided a reference for physical education teaching, improved the professional competence and quality of school physical education teachers, and formed the characteristics of school physical education and health curriculum.

Systematic and Durable Analysis of the Rule based Cauchy’s Inequality in solving the n-order Differential Equations

This paper instigates the multivariate generalization of the widely used Cauchy inequality 1 + x ≤ ex where x can be any non-negative real number. The results of this study can be the solution for the Cauchy’s problem for particular Ordinary Differential Equation (ODE). This is also related to study of the complete monotone function and the divided differences theory. The proof is based on the empty product convention notion and the Beppo Levi theorem of Monotone convergence. This study is also extended to multivariate generalization of the simultaneous inequalities. Key Words: ODEs, Inequalities, Population Dynamics, Simultaneous Inequalities, Divided differences.