Solutions Of Differential Equations Research Articles

Development of transient thermal-hydraulic analysis model for the two-phase loop thermosyphon

To comprehensively and accurately simulate and predict the flow and heat transfer characteristics of the two-phase loop thermosyphon (TPLT), a two-fluid two-phase model was developed. A six-equation set was formulated to account for two-phase flow, encompassing mass equations, momentum equations, and energy equations. The interactions between the fluids in terms of mass, momentum, and energy were also taken into account. To obtain numerical solutions for the partial differential equations within a staggered mesh framework, we employed the 2nd-order Lax-Wendroff scheme to obtain the discrete linear system, complemented by the inclusion of a flux limiter. The Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm was employed to address the solution of the linear equation system. Data from a small-scale TPLT test was used to assess the accuracy and reliability of both the physical models and the numerical method proposed in this study. Simulations were performed under two working conditions of the thermosyphon with filling ratio of 45 % and 64 %. The simulation results exhibited good agreement with the experimental data, demonstrating the accuracy of the model. Furthermore, the model is capable of providing information on the complex two-phase flow inside the TPLT that cannot be measured experimentally.

On a New Two-Point Taylor Expansion With Applications

A new two-point Taylor series expansion is proposed. The expansion is slightly different than the classical definition. The coefficients are calculated as recursive relations in a general form. The two-point Taylor expansion is applied to several functions which are odd, even, neither odd nor even. Functions having finite interval of convergence or infinite interval of convergence are investigated. The conditions for convergence are derived and the results are compared with the results of single-point Taylor expansions as well as two-point Taylor expansions reported in the literature. It is found that for a finite radius of convergence, two-point Taylor expansions can have a single convergence interval as well as two separate convergence intervals. Generally speaking, two-point Taylor expansions better represent the real function when the series is truncated. The new two-point expansion and the classical two-point expansion produced identical results for all the problems treated. Based on the results of this analysis, the asymmetric two-point Taylor expansion presented here does not have an advantage compared to the classical symmetric expansion. An application of the series to solution of a variable coefficient differential equation is also treated.

Analysis of solitary wave solutions in the fractional-order Kundu–Eckhaus system

The area of fractional partial differential equations has recently become prominent for its ability to accurately simulate complex physical events. The search for traveling wave solutions for fractional partial differential equations is a difficult task, which has led to the creation of numerous mathematical approaches to tackle this problem. The primary objective of this research work is to provide optical soliton solutions for the Frictional Kundu–Eckhaus equation (FKEe) by utilizing generalized coefficients. This strategy utilizes the Riccati–Bernoulli sub-ODE technique to effectively discover the most favorable traveling wave solutions for fractional partial differential equations. As a result, it facilitates the extraction of optical solitons and intricate wave solutions. The Backlund transformation is used to methodically construct a sequence of solutions for the specified equations. The study additionally showcases 3D and Density graphics that visually depict chosen solutions for certain parameter selections, hence improving the understanding of the outcomes.

Balanced implicit Patankar-Euler methods for positive solutions of stochastic differential equations of biological regulatory systems.

Stochastic differential equations (SDEs) are a powerful tool to model fluctuations and uncertainty in complex systems. Although numerical methods have been designed to simulate SDEs effectively, it is still problematic when numerical solutions may be negative, but application problems require positive simulations. To address this issue, we propose balanced implicit Patankar-Euler methods to ensure positive simulations of SDEs. Instead of considering the addition of balanced terms to explicit methods in existing balanced methods, we attempt the deletion of possible negative terms from the explicit methods to maintain positivity of numerical simulations. The designed balanced terms include negative-valued drift terms and potential negative diffusion terms. The proposed method successfully addresses the issue of divisions with very small denominators in our recently designed stochastic Patankar method. Stability analysis shows that the balanced implicit Patankar-Euler method has much better stability properties than our recently designed composite Patankar-Euler method. Four SDE systems are used to examine the effectiveness, accuracy, and convergence properties of balanced implicit Patankar-Euler methods. Numerical results suggest that the proposed balanced implicit Patankar-Euler method is an effective and efficient approach to ensure positive simulations when any appropriate stepsize is used in simulating SDEs of biological regulatory systems.

Implicit Block and Runge- Kutta type Methods for Solution of Second-Order Ordinary Differential Equations.

In this research, implicit discrete schemes which form our block integrators were developed for solving Initial Valued Problems of Ordinary Differential Equations. The equivalent second order Runge-Kutta type Methods (RKTM) were also constructed for the same purpose. Both methods were demonstrated on linear and nonlinear problems of Ordinary Differential Equations. Numerical results obtained from RKTM show that the method is competitive with the existing one.

FİNİTE ELEMENT ANALYSİS OF THE OUTER TUBE PART OF THE ENDOSCOPE

Even after cleaning and disinfecting the endoscope, the possibility of contamination is high and can adversely affect the health of patients. The US Centers for Disease Control report that there are approximately two million infected victims each year, and approximately 20,000 people succumb to the infection. In recent news, several deaths at UCLA Ronald Reagan Medical Center have been linked to bacterial contamination of endoscopes. Hence, there is a need to design an endoscope that will reduce or eliminate the chances of these deadly bacteria entering the body. In this article, we will look at the finite element analysis of the outer tube part of a newly designed disposable endoscope. The finite element method (SEM) is a popular method for the numerical solution of differential equations arising in engineering and mathematical modeling. Traditional areas of structural analysis include typical problem areas related to heat transfer, fluid flow, mass transport, and electromagnetic potential. Keywords: endoscope, outer tube, finite element method, mathematical modeling, analysis.

On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales

Abstract This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.

New solution of the non-linear Poisson-Boltzmann differential equation for solid particle dispersions in dissymmetrical electrolytes

A new mathematical solution to the non-linear Poisson-Boltzmann differential equation for solid-liquid dispersions in presence of different dissymmetrical electrolytes was given. The analytical expressions of the surface and charge density of solid particles were given. The variations of electrostatic potential ψ (x) and charge density σ (x) of dispersed particles against the distance x were obtained. For colloidal particles in presence of E(m-n) electrolytes with m≠n with m≥3,n≥3 and for E(2–3) and E(3-2) electrolytes, the mean electrostatic potential as a function of the distance was numerically integrated by Mathematica program version 13.The experimental study of silica suspensions in presence with the following electrolytes NaCl, Na2SO4, CaCl2, Na3PO4, AlCl3, Al2(SO4)3, Ca3(PO4)2, Na4P2O7 and Na5P3O10 led to confirm the theoretical predictions obtained from the analytical solution of Poisson-Boltzmann equation. The results obtained allowed to determine the surface potential as a function of pH of the suspension and the electrostatic potential versus the distance x. The variations of the dissociation coefficient of silica surfaces were determined. An important effect of the anion and cation valences of the dissymmetrical electrolytes on the surface charge density and potential was highlighted.

A note on Oishi’s lower bound for the smallest singular value of linearized Galerkin equations

Recently Oishi published a paper allowing lower bounds for the minimum singular value of coefficient matrices of linearized Galerkin equations, which in turn arise in the computation of periodic solutions of nonlinear delay differential equations with some smooth nonlinearity. The coefficient matrix of linearized Galerkin equations may be large, so the computation of a valid lower bound of the smallest singular value may be costly. Oishi’s method is based on the inverse of a small upper left principal submatrix, and subsequent computations use a Schur complement with small computational cost. In this note some assumptions are removed and the bounds improved. Furthermore a technique is derived to reduce the total computationally cost significantly allowing to treat infinite dimensional matrices.

An analytical approach for Yang transform on fractional-order heat and wave equation

A novel approach to locate the approximate analytical solutions for non-linear partial differential equations is presented in this paper: the Yang transformation method combined with the Caputo derivative. In the current work, we determine the fractional Heat and Wave equation’s approximate analytical solutions. This current work addresses the Yang transformation approach in addition with the Caputo derivative. The suggested method yields approximately analytical solutions in the form of series with a simple, straightforward mechanics and a proportionality dependent on values of the fractional-order derivative. A few numerical heat equation and wave equation problems are solved to show the usefulness and reliability of the method. The tabular form [tables 7–12] makes the claim that the absolute error decreased as the number of terms in the series increased. It is also confirmed that the results are graphical compatible.

Lower and upper bounds of the solution for the Lyapunov matrix differential equation and an application in input-output finite-time stability of linear systems

In control theory and practical engineering fields, such as the controllability, observability, input–output finite-time stability of the linear systems, it is a significant problem to study the properties of the solution for the Lyapunov matrix differential equation where there are no restrictions on the system matrix. In this paper, by constructing an equivalent form of the Lyapunov matrix differential equation and utilizing some important matrix eigenvalue inequalities, lower and upper bounds for the matrix solution of the Lyapunov matrix differential equation that remove the strict restrictions for the system matrix are proposed. As an application in linear systems, we show that our bounds can be used to discuss the input–output finite-time stability for linear systems. Finally, we give some corresponding numerical examples to illustrate the effectiveness and superiority of our results.

Analytical study of micropolar thermoelastic rectangular plate under three theories

A three-dimensional micropolar thermoelastic problem is considered for an isotropic rectangular plate that is subjected to simply support and isothermal on its four adjacent edges. Normal mode analysis is applied to the governing equations to form a vector-matrix differential equation and the analytical solution of the vector-matrix differential equation is obtained by using eigenvalue approach methodology. Numerical results are presented graphically along with theoretical analysis for sinusoidal variation of the stresses applied on the top surface of the rectangular plate.

The Existence and Uniqueness Conditions for Solving Neutrosophic Differential Equations and Its Consequence on Optimal Order Quantity Strategy

Background: Neutrosophic logic explicitly quantifies indeterminacy while also maintaining the independence of truth, indeterminacy, and falsity membership functions. This characteristic assumes an imperative part in circumstances, where dealing with contradictory or insufficient data is a necessity. The exploration of differential equations within the context of uncertainty has emerged as an evolving area of research. Methods: the solvability conditions for the first-order linear neutrosophic differential equation are proposed in this study. This study also demonstrates both the existence and uniqueness of a solution to the neutrosophic differential equation, followed by a concise expression of the solution using generalized neutrosophic derivative. As an application of the first-order neutrosophic differential equation, we discussed an economic lot sizing model in a neutrosophic environment. Results: This study finds the conditions for the existing solution of a first-order neutrosophic differential equation. Through the numerical simulation, this study also finds that the neutrosophic differential equation approach is much better for handling uncertainty involved in inventory control problems. Conclusions: This article serves as an introductory exploration of differential equation principles and their application within a neutrosophic environment. This approach can be used in any operation research or decision-making scenarios to remove uncertainty and attain better outcomes.

Existence and multiplicity of solutions of fractional differential equations on infinite intervals

In this research, we investigate the existence and multiplicity of solutions for fractional differential equations on infinite intervals. By using monotone iteration, we identify two solutions, and the multiplicity of solutions is demonstrated by the Leggett–Williams fixed point theorem.

Shear buckling of ship structures rectangular elements

In the work to determine the spectrum of critical loads and the corresponding forms of a rectangular clamped panel (plate) buckling under the action of balanced tangential forces on its contour, the Bubnov-Galerkin method using polynomials in two coordinates is proposed. This problem of the ship skin element pure shift does not have an exact closed solution, and the known approximate solutions require an analysis of their accuracy and reliability. The aim of the work is to obtain and analyze analytical solutions using polynomials of various degrees. Approximating deflection functions satisfying all the boundary conditions of the problem are represented sequentially by polynomials of 10th, 12th, 14th, 16th and 18th degrees in two coordinates with undefined coefficients. The solution of the main differential equation of the problem is found approximately in the integral sense, as a result of which homogeneous systems of linear algebraic equations with respect to unknown coefficients of polynomials are obtained. These systems contain a shear load as a parameter. To obtain eigenvalues (critical loads), the determinants of the systems are equated to zero. Numerical results are obtained in the Maple analytical computing system. For each approximation (polynomial), a power equation with respect to the critical load, the solution of which is paired values differing in signs is obtained. The forms of buckling are oblique waves. For a ship skin square panel, the first form of buckling is a single bulge along the diagonal of the panel. The second form is obtained in the form of two bulges directed in opposite directions (symmetrically-antisymmetrically with respect to the diagonals), etc. The numerical results are compared with the results of other authors. It is established that with an increase in the number of the polynomial terms, the initial critical loads and forms of buckling are specified, first of all.

A new approach to the fractional Abel k−integral equations and linear fractional differential equations in a fuzzy environment

This paper introduces some recent approaches to the fuzzy fractional Abel integral equations with respect to another function, which is called the fuzzy fractional Abel k−integral equations. The problem proposed here allows for the interpolation of different types of fuzzy classical fractional Abel integral equations, and the solvability of each type of integral equation is also discussed. Additionally, a new approach to the linear fuzzy fractional differential equations in the space of generalized fuzzy functions is introduced. Unlike previous approaches with gH-differentiability, this approach avoids making any prior assumptions about the solution's monotonic behavior during the solving process. It also overcomes the problem of multiple solutions for fuzzy fractional differential equations. To demonstrate the benefits of this approach, two initial value problems of linear fuzzy differential equations are compared using two different concepts of fractional derivatives: the Caputo fractional gH-differentiability and the Caputo fractional derivative in the generalized fuzzy space. Furthermore, some applications are provided in a viscoelastic material model and a population growth with harvesting model.

A note on the mean-square solution of the hypergeometric differential equation with uncertainties

The Fröbenius method of power series has been applied to several linear random differential equations. The interest relies on the derivation of a closed-form mean-square solution and on the possibility of approximating statistical measures at exponential convergence rate. In this paper, we deal with the hypergeometric differential equation with random coefficients and initial conditions. On the interval (0, 1), random power series centered at the regular singular point 0 are employed, which are given in terms of the hypergeometric function. We find the stochastic basis of mean-square solutions and solve random initial-value problems. The approximation of the expectation and the variance is studied and illustrated computationally.

New Conditions for Testing the Oscillation of Solutions of Second-Order Nonlinear Differential Equations with Damped Term

This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the studied equation without additional restrictions. Our approach improves the standard integral averaging technique to obtain simpler oscillation theorems for new classes of nonlinear differential equations. Two examples are presented to illustrate the importance of our findings.

An analysis on asymptotic stability of Hilfer fractional stochastic evolution equations with infinite delay

This article deals with the existence and asymptotic stability in the p-th moment of a mild solution for Hilfer fractional stochastic delay differential equations in Hilbert spaces. Our main results are obtained by fractional calculus, semigroup theory, stochastic analysis, and the Banach fixed point theorem. Further, a new set of sufficient conditions for existence and asymptotic stability in the p-th moment is derived with the help of a fixed point technique. Additionally, we provide an example to demonstrate the reliability of our results.

Stability in nonlinear integro-differential equations with functional delay on Time scale

The main purpose of this work is to show the stability of the zero solution for a non-linear neutral differential equation on space called the Time scale. In order to achieve our purposes, we assume a time scale , that is unbounded above and below, and we employ the contractive mapping theorem by converting the neutral differential equation into an equivalent integral equation, which allows us to reach the desired result of the asymptotic stability of the zero solution on Time scale provided that