Solutions Of Differential Equations Research Articles

Numerical solution of steady nonlinear differential equations for compressible flow through a spinning convergent divergent nozzle

Many projectiles tend to spin about their longitudinal axis while progressing in the forward direction. It helps in providing stability and a reference direction for guidance during their run. Many different projectiles employ a supersonic convergent-divergent nozzle to produce thrust for their forward motion; hence, the nozzle and overall whole propulsion system tend to spin about its axis of rotation. The main aim of this study is to observe the effect of spin on the nozzle. In this research, a converging bell-shaped diverging nozzle is numerically designed using a method of characteristics (MOC) for exit Mach number 3.21. Viscous simulations are performed for both two- and three-dimensional cases. The analysis is then performed with nozzle spinning about its axis of symmetry with a constant angular velocity of 10 revolutions per second. The analysis is repeated for the value of constant angular velocities to be 15 and 20 revolutions per second, and the behavior of flow with increasing angular velocity is examined. It has been observed that the exit Mach number and velocity decrease due to the radial protrusion of the boundary layer, and it has a negative impact on the performance of the nozzle. Moreover, the decrease of exit Mach number is in direct relation to increasing angular velocity.

Exponential stability results for second-order impulsive neutral stochastic differential equations

This paper is aimed to discuss the existence and exponential stability of mild solutions for second-order impulsive neutral stochastic differential equations (INSDEs). Firstly, we derive the existence of mild solutions for INSDEs by using the Banach fixed point theorem. Then, the exponential stability in the pth moment of mild solution for the considered INSDEs is proved by means of the generalized impulsive-integral inequality, which can effectively improve some known existing ones. Finally, as an application, an example is given to illustrate the efficiency of the obtained theoretical results.

On a computable Skorokhod's integral‐based estimator of the drift parameter in fractional SDE

AbstractThis paper deals with a Skorokhod's integral‐based least squares‐ (LS) type estimator of the drift parameter computed from multiple (possibly dependent) copies of the solution of a stochastic differential equation (SDE) driven by a fractional Brownian motion of Hurst index . On the one hand, some convergence results are established on our LS estimator when . On the other hand, when , Skorokhod's integral‐based estimators cannot be computed from data, but in this paper some convergence results are established on a computable approximation of our LS estimator.

The Lie point symmetry criteria and formation of exact analytical solutions for Kairat-II equation: Paul-Painlevé approach

The renowned non-linear Kairat-II equation, which expresses the surface curves, is examined meticulously in this work. There has never been a study before that discussed conserved quantities, sensitivity analysis, Hamiltonian function, Lie symmetry invariance criteria, and invariant solutions of Kairat-II equation The Lie invariance criteria are taken into consideration by the symmetry generators. The suggested technique leads to in a four-dimensional Lie algebra, where the translation point symmetries in space and time correlate with the conservation of mass and the energy, respectively, and remaining point symmetries are dilation and scaling. Firstly, we use symmetry reduction of Lie subalgebras to obtain closed-form invariant solutions. In specific reduction cases, we transform the Kairat-II equation into a spectrum of non-linear ordinary differential equations, which have the benefit of being able to offer an extensive selection of closed-form solitary wave solutions. For this equation, the Cauchy problem cannot be solved by the inverse scattering transform, therefore, traveling wave exact solutions are generated using the analytical Paul-Painlevé approach. In both two and three dimensions, the graphical behavior of specific solutions is presented for specific quantities of physical factors of examined equation. Utilizing conservation laws multipliers, the study culminates by determining an extensive set of local conservation laws for the non-linear Kairat-II equation that are applicable to arbitrary constant coefficients. Instead of focusing on the physical aspects of conservation laws, this study computed the conserved quantities using a mathematical perspective that can be used to identify potential symmetries of partial differential equations, guiding readers toward the solutions of partial differential equations. The existence of the Hamiltonian function is presented. The model’s sensitivity to various starting conditions is highlighted by the sensitivity analysis.

On the stability of periodic solutions of piecewise smooth periodic differential equations

On the stability of periodic solutions of piecewise smooth periodic differential equations

Plane Stress Problems for Isotropic Incompressible Hyperelastic Materials

The analysis of plane stress problems has long been a topic of interest in linear elasticity. The corresponding problem for non-linearly elastic materials is considered here within the context of homogeneous incompressible isotropic elasticity. It is shown that when the problem is posed in terms of the Cauchy stress, a semi-inverse approach must be employed to obtain the displacement of a typical particle. If however the general plane stress problem is formulated in terms of the Piola-Kirchhoff stress, the deformation of a particle requires the solution of a non-linear partial differential equation for both simple tension and simple shear, the trivial solution of which yields a homogeneous deformation. It is also shown that the general plane stress problem can be solved for the special case of the neo-Hookean material.

Solving Nonlinear Time-Fractional Partial Differential Equations Using Conformable Fractional Reduced Differential Transform with Adomian Decomposition Method

In this article, we use a new technique called conformable fractional reduced differential transform (CFRDT) with Adomian decomposition to estimate the solution of one and two-dimensional time-fractional partial linear and nonlinear differential equations with initial values. We explain the convergence analysis of this technique. The obtained results illustrate that the novel method is efficient and easy to use to find approximate solutions for the time-fractional partial differential equations (PDEs). Thus, the suggested method has a significant impact on how engineering, physics, and other disciplines solve fractional PDEs. Furthermore, we analyze the solution of problems with a 2D or 3D graphical representation by using Mathematica software.

Elzaki residual power series method to solve fractional diffusion equation.

The time-fractional order differential equations are used in many different contexts to analyse the integrated scientific phenomenon. Hence these equations are the point of interest of the researchers. In this work, the diffusion equation for a one-dimensional time-fractional order is solved using a combination of residual power series method with Elzaki transforms. The residual power series approach is a useful technique for finding approximate analytical solutions of fractional differential equations that needs the residual function's (n-1)α derivative. Since it is challenging to determine a function's fractional-order derivative, the traditional residual power series method's application is somewhat constrained. The Elzaki transform with residual power series method is an attempt to get over the limitations of the residual power series method. The obtained numerical solutions are compared with the exact solution of this equation to discuss the method's applicability and efficiency. The results are also graphically displayed to show how the fractional derivative influences the behaviour of the solutions to the suggested method.

Solving Linear and Nonlinear Fuzzy Fractional Volterra-Fredholm Integro Differential Equations Using Shehu Adomian Decomposition Method

Objectives: In applied sciences and engineering, fuzzy fractional differential equations (FFDEs) and fuzzy fractional integral equations (FFIEs) are a crucial topic. The main objective of this work is to discover an analytical approximate solution for the fuzzy fractional Volterra-Fredholm integro differential equations (FFVFIDE). In the Caputo concept, fractional derivatives are regarded. Methods: The Shehu transform is challenging to exist for nonlinear problems. So, the Shehu transform is combined with the Adomian decomposition method is called the Shehu Adomian decomposition method (SHADM) and has been proposed to solve both linear and nonlinear FFVFIDEs. Findings: Both linear and nonlinear FFVIFIDEs can be solved using this technique. For nonlinear terms, Adomian polynomials have been used. The main benefit of this approach is that it converges quickly to the exact solution. Figures and numerical examples demonstrate the expertise of the suggested approach. Novelty: The comparison between the exact solution and numerical solution is shown in figures for various values of fractional order . The numerical evolution demonstrates the efficiency and reliability of the proposed SHADM. The proposed approach is rapid, exact, and simple to apply and produce excellent outcomes. Keywords: Fractional calculus, fuzzy number, Mittag ­ Leffler function, Shehu Adomian decomposition method, fuzzy fractional Volterra­-Fredholm integro differential equation

Correlated dynamics of immune network and sl(3, R) symmetry algebra

Abstract We observed the existence of periodic orbits in immune network under transitive solvable Lie algebra. In this article, we focus to develop condition of maximal Lie algebra for immune network model and use that condition to construct a vector field of symmetry to study nonlinear pathogen model. We used two methods to obtain analytical structure of solution, namely normal generator and differential invariant function. Numerical simulation of analytical structure exhibits correlated periodic pattern growth under spatiotemporal symmetry, which is similar to the linear dynamical simulation result. We used Lie algebraic method to understand correlation between growth pattern and symmetry of dynamical system. We employ idea of using one parameter point group of transformation of variables under which linear manifold is retained. In procedure, we present the method of deriving Lie point symmetries, the calculation of the first integral and the invariant solution for the ordinary differential equation (ODE). We show the connection between symmetries and differential invariant solutions of the ODE. The analytical structure of the solution exhibits periodic behavior around attractor in local domain, same behavior obtained through dynamical analysis.

On the Solution of Blasius Boundary Layer Equations of Prandtl-Eyring Fluid Flow Past a Stretching Sheet

Objective: This paper investigates velocity profile for two-dimensional, incompressible, laminar forced convection flow of the fluid model for Prandtl-Eyring fluid past a stretching sheet in the presence of fluid parameters. Methods: The governing partial differential equation for the flow was transformed into non-linear ordinary differential equation by using the deductive one parameter group theoretic method and numerical solution of non-linear ordinary differential equation (ODE) is solved by MATLAB bvp4c solver. Findings: The solution of velocity profile obtained as a function of parameter and . The effect of the fluid parameter was discussed graphically. Novelty: The main goal of this article is to analyze boundary layer flow of Prandtl-Eyring fluid over a stretching surface. The conservation equations of mass, momentum are converted into non-linear ordinary differential equations along with boundary conditions using deductive one parameter group theoretic method and solved by MATLAB ODE solver. Comparisons with previously published works are made, and results show a high level of agreement. This type of research is applicable to extrusion, paper production, fiber glass production, hot rolling, condensation process, crystal growing, polymer sheets etc. Keywords: Boundary layer, laminar flow, Deductive one parameter Group theoretic method, Absolute invariant, Stretching Sheet, Prandtl-Eyring fluid

Tensor networks for solving the time-independent Boltzmann neutron transport equation

Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing in space, (ii) multigroup-in-energy, and (iii) discrete ordinates collocation in angle leads to large generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores. We then solve the tensorized generalized eigenvalue problem using a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying the method to two examples of 3D neutron transport problems, currently solved by the Los Alamos National Laboratory PARallel TIme-dependent SN (PARTISN) solver.1 We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to large compression. This allows for the storage of terrabyte-sized neutron angular flux eigenvectors in megabytes. Additionally, we create megabyte-sized full access TT representations of yottabyte-sized transport matrix operators. By leveraging the TT operators and solution methods, we obtain a 7500 times speedup when compared to the PARTISN solution time with an error of less than 10−5.

Existence of basic solutions of first order linear homogeneous set-valued differential equations

The paper presents various derivatives of set-valued mappings,their main properties and how they are related to each other.Next, we consider Cauchy problems with linear homogeneousset-valued differential equations with different types ofderivatives (Hukuhara derivative, PS-derivative andBG-derivative). It is known that such initial value problems withPS-derivative and BG-derivative have infinitely many solutions.Two of these solutions are called basic. These are solutions suchthat the diameter function of the solution section is amonotonically increasing (the first basic solution) or monotonicallydecreasing (the second basic solution) function. However, the secondbasic solution does not always exist. We provideconditions for the existence of basic solutions of such initialvalue problems. It is shown that their existence depends on thetype of derivative, the matrix of coefficients on the right-handand the type of the initial set. Model examples are considered.

Approximate solutions for fuzzy stochastic differential equations with Markovian switching

The fuzzy stochastic differential equations with Markovian switching are considered. First, under the Carath e ´ odory assumptions, the existence and uniqueness theorem for the aforementioned equations is given by means of stopping time techniques, discretization method, and the Gronwall-Bellman inequality. Then the boundedness of the second moment of the solutions for such equations is established. Furthermore, by using the averaging method and stochastic analysis, as well as the Bihari's inequality, this paper mainly demonstrates the asymptotic relationship between the solutions of the original equations and the corresponding averaged equations in terms of mean square and probability. Finally, the theoretical result of the averaging method is illustrated by an example.

Explicit solitary wave profiles and stability analysis of biomembranes and nerves

This paper examines the stability analysis and exact solitary wave solutions of the nonlinear partial differential equation known as the Heimburg model. The several types of solitary wave solutions, soliton solutions and Jacobi elliptic doubly periodic function solutions are explored by using the extended Sinh-Gordon equation expansion approach. These investigations exhibit the system’s astounding diversity of waveforms, highlighting its potential applications in nerves and biomembranes. By selecting some appropriate values for the parameters, 3D, 2D, and its corresponding contour graph are plotted to represent the physical relevance of some of the solutions. Additionally, the linearized stability of this system is analyzed. The suggested approach is the finest resource for the analytical investigation of any nonlinear issue that occurs in various scientific fields.

Exact solutions for nonlinear partial differential equations via a fusion of classical methods and innovative approaches

This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the form utt+a(x,t)uxt+b(t)ut=α(x,t)+G(u)(ut+a(x,t)ux)e-∫b(t)dt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{tt}+a(x,t)u_{xt}+b(t)u_{t}=\\alpha (x,t)+ G(u)(u_{t}+a(x,t)u_{x})e^{-\\int b(t)dt}$$\\end{document} and utm(utt+a(x,t)uxt)+b(t)utm+1=e-(m+1)∫b(t)dt(ut+a(x,t)ux)F(u,ute∫b(t)dt).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{t}^{m}(u_{tt}+a(x,t)u_{xt})+b(t)u_{t}^{m+1}=e^{-(m+1)\\int b(t)dt}(u_{t}+a(x,t)u_{x}) F(u,u_{t}e^{\\int b(t)dt}).$$\\end{document} We provide numerical validation through several examples to ensure accuracy and reliability. Our approach enhances the applicability of analytical solution methods for a broader range of NLPDEs.

A differential equation for inverse point kinetics

The reactivity that represents the percentage departure of a nuclear reactor from its criticality condition is perhaps the most important individual property since it is directly related to the nuclear reactor’s safe operation. Control systems are used to limit the rate of increase in reactivity. The development of systems for monitoring the reactivity, from the nuclear power data acquired from the plant's onboard electronics, requires a constant improvement of the nuclear kinetic models.The transient situations that emerge in nuclear reactors can be predicted from the changes in the neutron flux and, therefore, it is possible to make a sufficiently accurate prediction about the consequences of the disturbances relating to the magnitude of the neutron flux, which varies over time, by using the integral parameters that represent the nuclear reactor core. The point kinetic equations relate these parameters and, in this way, allow the transient situations that can occur in a nuclear reactor to be studied.The reactivity is usually determined by solving the inverse point kinetic equation, which is an integral equation that depends on the nuclear power variation history. In this paper, we present a method to calculate the reactivity through the solution of an ordinary differential equation that does not depend on the nuclear power variation history, then allows the prompt recovery of the reactimeter in case of loss of the nuclear power data acquisition or malfunction of such equipment (Zarei, 2022; Ansari, 1991).The presented paper is organized as follows: Section 1 contains the basic concepts of point kinetics equations and the integral formalism for the reactivity calculation. The mathematical formalism that leads to the differential equation for the reactivity is presented in Section 2. The prompt recovery of the reactivity using the proposed method is discussed in Section 3. The results are shown in Section 4 and the conclusion is presented in Section 5.

A higher-order approximation method for jump-diffusion SDEs with a discontinuous drift coefficient

We present the first higher-order approximation scheme for solutions of jump-diffusion stochastic differential equations with discontinuous drift. For this transformation-based jump-adapted quasi-Milstein scheme, we prove Lp-convergence order 3/4. To obtain this result, we prove that under slightly stronger assumptions (but still weaker than assumptions applied before) a related jump-adapted quasi-Milstein scheme has convergence order 3/4; in a special case it even has convergence order 1. Order 3/4 is conjectured to be optimal.

Travelling Wave Solutions for Some Time-Fractional Nonlinear Differential Equations

This study employs the powerful generalized Kudryashov method to address the challenges posed by fractional differential equations in mathematical physics. The main objective is to obtain new exact solutions for three important equations: the (3+1)-dimensional time fractional Jimbo-Miwa equation, the (3+1)-dimensional time fractional modified KdV-Zakharov-Kuznetsov equation, and the (2+1)-dimensional time fractional Drinfeld-Sokolov-Satsuma-Hirota equation. The generalized Kudryashov method is highly versatile and effective in addressing nonlinear problems, making it a pivotal component in our research. Its adaptability makes it useful in diverse scientific disciplines. The method simplifies complex equations, improving our analytical capabilities and deepening our understanding of system dynamics. Additionally, we define fractional derivatives using the conformable fractional derivative framework, providing a strong foundation for our mathematical investigations. This paper examines the effectiveness of the generalized Kudryashov method in solving complex challenges presented by fractional differential equations and aims to provide guidance for future studies.

Numerical solution of fractional PDEs through wavelet approach

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1]$$\\end{document} is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof.