Terms Of Differential Equations Research Articles

Solving second‐order differential equations in terms of confluent Heun's functions

This paper presents an algorithm that checks if a given second‐order homogeneous linear differential equation can be reduced to the confluent Heun's equation by using the change of variables transformation and the exp‐product transformation. The main purpose of this paper is finding solutions in terms of the confluent Heun's functions for the given differential equation.

Analytic solution to functional differential equations via Bell’s polynomials

Abstract It is shown how to approximate the solution of functional differential equations in terms of Bell’s polynomials. Some numerical checks are shown, by using the computer algebra system Mathematica © {{}^{\copyright}} .

Expressing Solutions of Bessel Differential Equation in terms of Hypergeometric Functions

The BDE (Bessel differential equation) is a second-order linear ordinary differential equation (ODE), and it is considered one of the most significant differential equations because of its extensive applications. The solutions of this differential equation are called Bessel functions. These solutions can be expressed in terms of hypergeometric functions, and in this study, the Bessel functions of the first kind are worked. Hypergeometric functions are the sum of a hypergeometric series. A series ∑▒u_n is known as hypergeometric when the ratio u_(n+1)⁄u_n is a rational function of n. BDE has coefficients with variables; therefore, it is solved by the Frobenius approach. We implement some mathematical steps based on the definition of hypergeometric functions to express the solutions in terms of hypergeometric function. The result shows that the solution of the BDE in terms of hypergeometric function is f(x)=a_0 x^p (_1^ )F_2 (1;1+(p+ρ)/2,1+(p-ρ)/2;x^2⁄4) and the two linearly independent solutions for the BDE of order ρ in terms of hypergeometric function are 〖f_1 (x)=a〗_0 x^ρ (_0^ )F_1 (1+ρ;x^2⁄4) and〖〖 f〗_2 (x)=a〗_0 x^ρ (_0^ )F_1 (1-ρ;x^2⁄4).

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators

We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation’s solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties.

Significance of Cu-Fe3O4 on fractional Maxwell fluid flow over a cone with Newtonian heating

The goal of this research is to investigate fractional Maxwell hybrid nanofluids utilizing partial differential equations in terms of Caputo time fractional derivatives. Specifically, the effect of Newtonian heating on the thermal performance of a fractional Maxwell hybrid nanofluid moving over a permeable cone in the presence of thermal radiation and heat generation is considered. The Crank–Nicolson method and L1 algorithmt of Caputo derivative are used to find numerical solutions to the considered nonlinear problem. The effects of significant flow factors on fluid properties are examined and illustrated in various graphs. According to the results, the thermal performance of the fluid raised by 0.4%, 6.1%, and 3.1% on adding 4% volume fraction of Fe3O4, Cu, and Cu−Fe3O4, respectively, in the base fluid.

On Hamiltonian structures of quasi-Painlevé equations

Abstract We describe the quasi-Painlevé property of a system of ordinary differential equations in terms of a global Hamiltonian structure on an analogue of Okamoto’s space of initial conditions for the Painlevé equations. In the quasi-Painlevé case, the Hamiltonian structure is with respect to a two-form which is allowed to have certain zeroes on the surfaces forming the space of initial conditions, as opposed to holomorphic symplectic forms in the case of the Painlevé equations. We provide the spaces and Hamiltonian structures for several known quasi-Painlevé equations and also for a new example, which we prove to have the quasi-Painlevé property via the Hamiltonian structure and construction of an appropriate auxiliary function which remains bounded on solutions.

Refined beam finite element model for thin-walled multi-cell box girders considering distortion and secondary distortional moment deformation effect

In this paper, a novel one-dimensional refined beam finite element model (FEM) accounting for non-uniform distortional warping and secondary distortional moment deformation effect (SDMDE) is formulated for the distortional analysis of thin-walled multi-cell box girders with cantilevered flanges. The extra distortional angle is introduced to incorporate SDMDE, and the validity of which is proved through rigorous theoretical derivation. The in-plane and out-plane deformations at any point around the edge of transverse section built upon the distortional center are unified in the same form and described in detail. The governing differential equations in terms of two generalized displacements, distortional angle χ and primary distortional angle curvature Θ, are derived based on the principle of minimum potential energy (MPE) and solved exactly so as to provide non-polynomial shape functions for the establishment of element stiffness matrix and nodal load vector. A series of multi-cell rectangular hollow section (RHS) girders are analyzed and particular extensions are given to single-cell and multi-cell bridge girders with cantilevered flanges. Comparisons to the results generated from corresponding ones provided by pioneering work or shell FEM analyses not only validate the correctness and reliability of the present beam element formulations but also highlight the significance of the inclusion of SDMDE when the cantilevered flanges are arranged.

Alcubierre warp drive in Bohmian Quantum Gravity

In the realm of theoretical physics, the Alcubierre warp drive metric has long been a topic of fascination. However, the feasibility of this seemingly miraculous method of faster-than-light travel has been limited by the requirement of exotic matter, which has proven elusive to obtain. This research has revealed a potential breakthrough in the quest for exotic matter, through the coupling of the Alcubierre warp drive metric to a quantum mechanical scalar matter field. By mapping the requirement of exotic matter into a conformal wave equation, it becomes possible to express the problem as a fourth-order partial differential equation in terms of the quantum mechanical density. This research reveals that the key to solving the exotic matter problem lies in finding a proper quantum mechanical density that satisfies the proposed partial differential equation.

Finite elements simulation MHD free convection in a rectangular embedded corrugated rods cavity filled with CuO/water nanofluid

The convective heat transfer and fluid is investigated in a rectangle chamber along with corrugated heated rods. For preferable novelties, this study considered porous media, micropolar fluid and an inclined magnetic field effect. The physical illustrations of nanofluid flow is assimilated in terms of differential equations. We use the finite element approach to tackle the fluid flow and heat transfer in rectangular chamber. Flow regulating parameters are Rayleigh number, Darcy number and Hartman number. The horizontal and vertical velocities and temperature distribution are provided through the use of contour plots and line graphs. The results show that heat transmission rates increase due to the increasing values of Hartmann number. Because Lorentz forces are resistive, the velocity profile tends to flatten down as long as Hartman number increases. The Nusselt number and velocity profile have greater values around the heat boundaries.

The Solution of a System of Higher-Order Difference Equations in Terms of Balancing Numbers

In this paper, we are interested in the closed-form solution of the following system of nonlinear difference equations of higher order,un+1 = 1/34-vn-m , vn+1 = 1/34-un-m, n, m ∈ N0,and the initial values u-j and v-j , j∈{0, 1, ..., m} are real numbers do not equal 34. We show that the solutions of this system are associated with Balancing numbers. As consequence, these solutions are also associated with Pell numbers, Pell-Lucas numbers, and Lucas-Balancing numbers. It is shown that the global stability of positive solutions of this system holds. Our results are illustrated via numerical examples.

Analysis of linearized elasticity models with point sources in weighted Sobolev spaces: applications in tissue contraction

In order to model the contractive forces exerted by fibroblast cells in dermal tissue, we propose and analyze two modeling approaches under the assumption of linearized elasticity. The first approach introduces a collection of point forces on the boundary of the fibroblast whereas the second approach employs an isotropic stress point source in its center. We analyze the resulting partial differential equations in terms of weighted Sobolev spaces and identify the singular behavior of the respective solutions. Two finite element method approaches are proposed, one based on a direct application and another in which the singularity is subtracted and a correction field is computed. Finally, we confirm the validity of the modeling approach, demonstrate convergence of the numerical methods, and verify the analysis through the use of numerical experiments.

Numerical approach for temperature dependent properties of sutterby fluid flow with induced magnetic field past a stretching cylinder

In this analysis, Sutterby fluid flow of induced magnetic hydrodynamic at a stretchable cylinder is deliberated. The viscous dissipation impacts with temperature-dependent properties are applied on the fluid flow. The governing equations are developed using boundary layer approximation in terms of differential equations. The differential equations are dimensionless by means of suitable transformation. The dimensionless system is elucidated by the numerical procedure. Governing physical parameters influences are offered through graphs as well as tabular form. The curves of the velocity function show to enhance due to an increment in the decay-resistant parameter. The curves of the velocity function increased due to higher values of the magnetic field parameter. The sponginess parameter and velocity have inverse relation presented in graph. Temperature augmented due to increasing values of ε. The variable thermal conductivity and temperature have direct relation. Due to this relation, temperature boosted up by boosting the thermal conductivity.

On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers

ABSTRACT In this paper, we represent that the following three-dimensional system of difference equations x n + 1 = α y n + a y n y n − β z n − 1 , y n + 1 = β z n + b z n z n − γ x n − 1 , z n + 1 = γ x n + c x n x n − α y n − 1 , n ∈ ℕ 0 , $$\matrix{{{x_{n + 1}} = \alpha {y_n} + {{a{y_n}} \over {{y_n} - \beta {z_{n - 1}}}},\quad {y_{n + 1}} = \beta {z_n} + {{b{z_n}} \over {{z_n} - \gamma {x_{n - 1}}}},\quad {z_{n + 1}} = \gamma {x_n} + {{c{x_n}} \over {{x_n} - \alpha {y_{n - 1}}}},\qquad n \in {{\mathbb N}_0},} \cr} $$ where the parameters a, b, c, α, β, γ and the initial values x −i , y −i , z −i , i ∈ {0, 1}, are real numbers, can be solved in closed form by using transformation. We analyzed the solutions in 10 different cases depending on whether the parameters are zero or nonzero. It is noteworthy to depict that the solutions of some particular cases of this system are presented in terms of generalized Fibonacci numbers. Note that our results considerably extend and improve some recent results in the literature.

Green’s function for thick laminated composite plates with coupled stretching-bending and transverse shear deformation

In this paper, the Green’s function of a thick laminated composite plate is derived for the first time in the literature. The plate can be unsymmetric with respect to the mid-plane such that the coupling of in-plane stretching and out-of-plane bending deformations occurs. Also, the plate is thick enough that the transverse shear deformation cannot be ignored. Through the plane wave decomposition method, the partial differential governing equations based upon the first order shear deformation plate theory can be reorganized into the ordinary differential equations in terms of the transformed variable. With the governing equations reorganized into matrix form, the Green’s functions are expressed in terms of matrix exponential. Through eigen-decomposition, the matrix exponential including its related integrals can be evaluated explicitly. The explicit solutions of Green’s functions in transformed domain are obtained accordingly. In order to improve the accuracy and efficiency of numerical integration for the required transform integrals, a special quadrature rule which can eliminate the integral singularity is suggested. To have a fair verification and to extend the applicability of newly derived Green’s functions, the associated boundary element method is also introduced in this study.

Thermal case study of nanofluid flow in partially heated rectangular enclosure rooted with sinusoidal heated rods and inclined magnetic field

The natural convection nanofluid flow in a partially heated rectangular enclosure is studied numerically. For better novelty, an inclined magnetic field is carried externally. The wavy heated rods are installed at both the bottom and upper walls of an enclosure. Copper-Oxide (CuO) nanoparticles are suspended in the flow field. The nanofluid flow is mathematically formulated in terms of differential equations. The finite element method is used to examine the flow field in a partially heated rectangular enclosure. The flow-affecting parameters are Hartmann number and angle of inclination. The quantity of interest includes horizontal velocity, vertical velocity, and Nusselt number. The outcomes are shared in terms of contour and line plots. It is observed that for increasing values of the Hartmann number, the Nusselt number at the upper heated sinusoidal rod shows a higher magnitude in comparison with a lower heated corrugated rod.

The Uses and Advantages of Kirchhoff’s Laws vs. Differential Equations in Pharmacology, Pharmacokinetics, and (Even) Chemistry

In chemistry, rate processes are defined in terms of rate constants, with units of time−1, and are derived by differential equations from amounts. In contrast, when considering drug concentrations in biological systems, particularly in humans, rate processes must be defined in terms of clearance, with units of volume/time, since biological volumes, which are highly dependent on drug partition into biological tissues, cannot be easily determined. In pharmacology, pharmacokinetics, and in making drug dosing decisions, drug clearance and changes in drug clearance are paramount. Clearance is defined as the amount of drug eliminated or moved divided by the exposure driving that elimination or movement. Historically, all clearance derivations in pharmacology and pharmacokinetics have been based on the use of differential equations in terms of rate constants and amounts, which are then converted into clearance equations when multiplied/divided by a hypothesized volume of distribution. Here, we show that except for iv bolus dosing, multiple volumes may be relevant. We have recently shown that clearance relationships, as well as rate constant relationships, may be derived independent of differential equations using Kirchhoff’s Laws from physics. Kirchhoff’s Laws may be simply translated to recognize that when two or more rate-defining processes operate in parallel, the total value of the overall reaction parameter is equal to the sum of those rate-defining processes. In contrast, when two or more rate-defining processes operate in series, the inverse of the total reaction parameter is equal to the sum of the inverse of those rate-defining steps.Graphical

An in-silico framework for modeling optimal control of neural systems.

Brain-machine interfaces have reached an unprecedented capacity to measure and drive activity in the brain, allowing restoration of impaired sensory, cognitive or motor function. Classical control theory is pushed to its limit when aiming to design control laws that are suitable for large-scale, complex neural systems. This work proposes a scalable, data-driven, unified approach to study brain-machine-environment interaction using established tools from dynamical systems, optimal control theory, and deep learning. To unify the methodology, we define the environment, neural system, and prosthesis in terms of differential equations with learnable parameters, which effectively reduce to recurrent neural networks in the discrete-time case. Drawing on tools from optimal control, we describe three ways to train the system: Direct optimization of an objective function, oracle-based learning, and reinforcement learning. These approaches are adapted to different assumptions about knowledge of system equations, linearity, differentiability, and observability. We apply the proposed framework to train an in-silico neural system to perform tasks in a linear and a nonlinear environment, namely particle stabilization and pole balancing. After training, this model is perturbed to simulate impairment of sensor and motor function. We show how a prosthetic controller can be trained to restore the behavior of the neural system under increasing levels of perturbation. We expect that the proposed framework will enable rapid and flexible synthesis of control algorithms for neural prostheses that reduce the need for in-vivo testing. We further highlight implications for sparse placement of prosthetic sensor and actuator components.

Linearisation of a second-order nonlinear ordinary differential equation

We analyse nonlinear second-order differential equations in terms of algebraic properties by reducing a nonlinear partial differential equation to a nonlinear second-order ordinary differential equation via the point symmetry f(v)∂v. The eight Lie point symmetries obtained for the second-order ordinary differential equation is of maximal number and a representation of the sl(3,R) algebra. We extend this analysis to a more general nonlinear second-order differential equation and we obtain similar interesting algebraic properties.

Stability of Anisotropy Pressure in Self-Gravitational Systems in f(G) Gravity

This investigation aims to explore certain variables which are considered responsible for generating pressure anisotropy in dynamical spherically symmetric stellar systems against the background of the stringy-inspired Gauss–Bonnet modification of general relativity. We explore the hydrostatic equilibrium of self-gravitating systems by taking into account the modified form of Tolman–Oppenheimer–Volkoff for the quadratic-f(G) gravitational model. In this respect, we formulate a differential equation in terms of the Weyl curvature scalar, also described as an evolution equation, which is essential for understanding the evolution of the stellar structure. Finally, we conclude that the existence of some fluid variables such as shear, heat flux and the irregular behavior of energy density in the presence of an extra degree f(G)-terms in the fluid flow that are the elements that cause anisotropy in the initially isotropic stellar structure. The comparison of the presented results with those of the classical model shows that they are physically relevant and compatible.