Variable Coefficient Ordinary Differential Equations Research Articles

Bifurcation analysis, and exact solutions of the two-mode Cahn–Allen equation by a novel variable coefficient auxiliary equation method

This document elaborates on a newly introduced analytical method known as the “Variable Coefficient Generalized Abel Equation Method,” as proposed by Hashemi in Hashemi (2024), designed specifically for addressing the two-mode Cahn–Allen equation. Diverging from conventional techniques that heavily rely on constant coefficient ordinary differential equations and auxiliary ordinary differential equations, our method innovatively incorporates variable coefficient ordinary differential equations within a sub-equation framework. Demonstrating its versatility, we apply this innovative technique to the two-mode Cahn–Allen equation, showcasing its effectiveness and efficiency through the derivation of analytical solutions. Notably, this method emerges as a promising tool for tackling complex nonlinear partial differential equations prevalent in fluid dynamics and wave propagation scenarios. Beyond merely expanding the repertoire of available analytical tools, our approach contributes to advancing solutions for various models within the realm of mathematical physics. Various forms of exact solutions, including exponential-type solutions, Kink solitons, dark solitons, and bright soliton solutions, are obtained for the model under consideration. Moreover, we delve into the analysis of bifurcation, chaotic behavior, and sensitivity within the context of the two-mode Cahn–Allen model, further enhancing the depth and breadth of our study. Three equilibria are analyzed across various classifications, including center point, focus point, saddle point, and node point. Chaotic behavior of the corresponding dynamical system is considered by adding the function ω1sin(ω2ζ). Lastly, sensitivity analysis of the system is conducted by examining different parameters of the model and imposing noise to the initial conditions.

Modified Deflection Theory for Preliminary Design of Self-Anchored Suspension Bridges

A modified deflection theory is developed for preliminary design of self-anchored suspension bridges. The proposed theory modifies the questionable approach of the existing theory considering the initial fabrication camber and overcomes the limitation that the hangers are assumed inextensible, which results in a stiffer bridge system and thus underestimation of the main cable and girder deflections. In addition, in order to avoid the inconvenience of solving a system of nonlinear equations iteratively for the preliminary design, the tower flexural stiffness is neglected rationally to obtain a system of linear equations only. With the aid of all force equilibrium and deformation compatibility conditions for the entire bridge system, the modified deflection theory is formulated. Its solution procedure is presented, which leads to a complicated sixth-order variable-coefficient ordinary differential equation, and a practical approximate solution to the equation is sought. To verify the proposed theory, a bridge example is investigated, and the results are compared to those from the previous deflection theory and complex finite element analysis. The comparisons demonstrate the effectiveness of the modified deflection theory.

Simulating single-particle dynamics in magnetized plasmas: The RMF code.

The RMF (Rotating Magnetic Field) code is designed to calculate the motion of a charged particle in a given electromagnetic field. It integrates Hamilton's equations in cylindrical coordinates using an adaptive predictor-corrector double-precision variable-coefficient ordinary differential equation solver for speed and accuracy. RMF has multiple capabilities for the field. Particle motion is initialized by specifying the position and velocity vectors. The six-dimensional state vector and derived quantities are saved as functions of time. A post-processing graphics code, XDRAW, is used on the stored output to plot up to 12 windows of any two quantities using different colors to denote successive time intervals. Multiple cases of RMF may be run in parallel and perform data mining on the results. Recent features are a synthetic diagnostic for simulating the observations of charge-exchange-neutral energy distributions and RF grids to explore a Fermi acceleration parallel to static magnetic fields.

Analytical modeling of variable thickness cylindrical shallow shells using extended Kantorovich method

The objective of the current paper is to provide novel numerical and analytical solutions for the variable cross section shallow shells under static pressure loads. First, the governing equations and the corresponded boundary conditions of the system are derived using the principle of virtual work. Then using the extended Kantorovich method (EKM), the partial differential equations of the system are transformed into two sets of variable coefficient ordinary differential equations, which are solved iteratively using both a novel numerical finite difference scheme, as well as an analytical perturbation tool. The convergence of the developed finite difference scheme is analyzed and it is observed that considering 600 nodes in the normalized domain is sufficient for achieving highly accurate results. Similarly, analytical simulations reveal that a fourth order perturbation expansion leads to an error of less than 0.2% in calculating the pick transverse response. The convergence of the EKM is also investigated and it is observed that the maximum transverse displacement can be predicted with less than 0.01% error using only two iterates. Additionally, variations of the stress and strain components in the mid-surface are studied using contour plots. Finally, parametric studies are carried out to characterize the effect of the shell's aspect ratio on the maximum transverse deflection. The analytical and numerical findings reported in this paper are closely validated with FE results as well as other available data in the literature.

Lateral-Torsional Stability of Rectangular Prismatic Beams Using Some Analytical Approximation Techniques

The paper presents simple computational algorithms for analyzing the lateral-torsional buckling of prismatic beams with rectangular cross-sections under bending action due to uniform and nonuniform loads by the Adomian decomposition method (ADM) and variational iteration method (VIM). Unlike the numerical techniques that lead to a discretization process, the proposed method allows us to derive the solution in terms of an analytical function for the problem considered. Although the governing equations of the problem appear as a system of two coupled variable coefficient ordinary differential equations, they reduce to a single equation for rectangular beams. The buckling loads for different loading conditions are computed, with the results for the simple beam compared with previous available results by the differential transformation method (DTM), variational iteration method (VIM) and finite element method (FEM) based on coupled governing equations. The results clearly show the efficiency and advantage of the present technique over those based on the coupled governing equations using the DTM and VIM in view of the number of terms required to obtain the convergent solution.

Implementation of Detailed Chemistry Mechanisms in Engine Simulations

The stiffness of large chemistry mechanisms has been proved to be a major hurdle toward predictive engine simulations. As a result, detailed chemistry mechanisms with a few thousand species need to be reduced based on target conditions so that they can be accommodated within the available computational resources. The computational cost of simulations typically increases super-linearly with the number of species and reactions. This work aims to bring detailed chemistry mechanisms within the realm of engine simulations by coupling the framework of unsteady flamelets and fast chemistry solvers. A previously developed tabulated flamelet model (TFM) framework for nonpremixed combustion was used in this study. The flamelet solver consists of the traditional operator-splitting scheme with variable coefficient ordinary differential equation (ODE) solver (VODE) and a numerical Jacobian for solving the chemistry. In order to use detailed mechanisms with thousands of species, a new framework with the Livermore solver for ODEs in sparse form (LSODES) chemistry solver and an analytical Jacobian was implemented in this work. Results from 1D simulations show that with the new framework, the computational cost is linearly proportional to the number of species in a given chemistry mechanism. As a result, the new framework is 2–3 orders of magnitude faster than the conventional variable coefficient ODE (VODE) solver for large chemistry mechanisms. This new framework was used to generate unsteady flamelet libraries for n-dodecane using a detailed chemistry mechanism with 2755 species and 11,173 reactions. The engine combustion network (ECN) spray A experiments, which consist of an igniting n-dodecane spray in turbulent, high-pressure engine conditions are simulated using large eddy simulations (LES) coupled with detailed mechanisms. A grid with 0.06 mm minimum cell size and 22 ×106 peak cell count was implemented. The framework is validated across a range of ambient temperatures against ignition delay and liftoff lengths (LOLs). Qualitative results from the simulations were compared against experimental OH and CH2O planar laser-induced fluorescence (PLIF) data. The models are able to capture the spatial and temporal trends in species compared to those observed in the experiments. Quantitative and qualitative comparisons between the predictions of the reduced and detailed mechanisms are presented in detail. The main goal of this study is to demonstrate that detailed reaction mechanisms (∼1000 species) can now be used in engine simulations with a linear increase in computation cost with number of species during the tabulation process and a small increase in the 3D simulation cost.

New Similarity Solutions of a Generalized Variable-Coefficient Gardner Equation with Forcing Term

The compatibility method is used for a generalized variable-coefficient Gardner equation (GVGE) with a forcing term. By the compatibility of the considered equation and a non-classical symmetry of a given form, four types of symmetry are obtained. Then, by solving the characteristic equations of symmetry, the GVGE is reduced to variable coefficients ordinary differential equations, and rich varieties of new similarity solutions are presented. Our results show that the compatibility method can be employed for variable coefficients nonlinear evolution equations with forcing terms.

Nonlinear Transient Response of Functionally Graded Material Sandwich Doubly Curved Shallow Shell Using New Displacement Field

In this paper, the nonlinear transient dynamic response of functionally graded material (FGM) sandwich doubly curved shell with homogenous isotropic material core and functionally graded face sheet is analyzed using a new displacement field on the basis of Reddy’s third-order shear theory for the first time. The equivalent material properties for the FGM face sheet are assumed to obey the rule of simple power law function in the thickness direction. Based on Reddy’s theory of higher shear deformation, a new displacement field is developed by introducing the secant function into transverse displacement. Four coupled nonlinear differential equations are obtained by applying Hamilton’s principle and Galerkin method. It is assumed that the FGM sandwich doubly curved shell is subjected to step loading, air-blast loading, triangular loading, and sinusoidal loading, respectively. On the basis of double-precision variable-coefficient ordinary differential equation solver, a new program code in FORTRAN software is developed to solve the nonlinear transient dynamics of the system. The influences of core thickness, volume fraction, core-to-face sheet thickness ratio, width-to-thickness ratio and blast type on the transient response of the shell are discussed in detail through numerical simulation.

Extending the Constant Coefficient Solution Technique to Variable Coefficient Ordinary Differential Equations

We introduce a class of second-order ordinary differential equations (ODEs) with variable coefficients whose closed-form solutions can be obtained by the same method used to solve ODEs with constant coefficients. General solutions for the homogeneous case are discussed.

Spatially Dispersionless, Unconditionally Stable FC–AD Solvers for Variable-Coefficient PDEs

We present fast, spatially dispersionless and unconditionally stable high-order solvers for partial differential equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain “Fourier continuation” (FC) method for the resolution of the Gibbs phenomenon on equi-spaced Cartesian grids, together with (ii) A new, preconditioned, FC-based solver for two-point boundary value problems for variable-coefficient Ordinary Differential Equations, and (iii) An Alternating Direction strategy, generalize significantly a class of FC-based solvers introduced recently for constant-coefficient PDEs. The present algorithms, which are applicable, with high-order accuracy, to variable-coefficient elliptic, parabolic and hyperbolic PDEs in general domains with smooth boundaries, are unconditionally stable, do not suffer from spatial numerical dispersion, and they run at Fast Fourier Transform speeds. The accuracy, efficiency and overall capabilities of our methods are demonstrated by means of applications to challenging problems of diffusion and wave propagation in heterogeneous media.

NUMERICAL TOOLS FOR THREE DIMENSIONAL SIMULATIONS OF THE ROTATING DETONATION ENGINE IN COMPLEX GEOMETRIES

This paper describes the development of a computational code REFLOPS USG (REactive FLOw solver for Propulsion Systems on UnStructured Grids) based on the Favre averaged Navier-Stokes equations with chemical reactions for semi-ideal multicomponent gas to predict the structure and dynamics of three-dimensional unsteady detonation as it occurs in the Rotating Detonation Engine (RDE). This work provides an overview of second order accurate in time and space finite volume method applied to conservation equations and its implementation on unstructured self-adaptive tetrahedral or hexahedral three-dimensional cell-centred meshes. The inviscid fluxes are given by the Riemann solver and stabilization is ensured by the proper limiters inherited from the TVD theory or gradient based limiters. The stiff equations of chemical kinetics are solved by use of implicit DVODE (Double precision Variablecoefficient Ordinary Differential Equation solver, with fixed-leading-coefficient implementation) routine or by explicit Chemeq2 routine. Additional improvements are incorporated into the code such as parallelization in OpenMP and implementation of NVIDIA CUDA technology. REFLOPS USG has become a fundamental numerical tool in the research of RDE at the Institute of Aviation in Warsaw, in frame of Innovative Economy project UDA-POIG.01.03.01-14-071 ‘Turbine engine with detonation combustion chamber’ supported by EU and Ministry of Regional Development, Poland. The simulations presented in this paper are based on inviscid or viscous multicomponent semi-ideal gas flow with chemical reactions. Due to high computational costs only simple chemical reaction mechanisms are used here.

Comparison of approximate symmetry and approximate homotopy symmetry to the Cahn–Hilliard equation

Complete infinite order approximate symmetry and approximate homotopy symmetry classifications of the Cahn–Hilliard equation are performed and the reductions are constructed by an optimal system of one-dimensional subalgebras. Zero order similarity reduced equations are nonlinear ordinary differential equations while higher order similarity solutions can be obtained by solving linear variable coefficient ordinary differential equations. The relationship between two methods for different order are studied and the results show that the approximate homotopy symmetry method is more effective to control the convergence of series solutions than the approximate symmetry one.

An indirect variable transformation approach and Jacobi elliptic solutions to Korteweg de Vries equation

Based on a variable change and the variable separated ODE method, an indirect variable transformation approach is proposed to search exact solutions to special types of partial differential equations (PDEs). The new method provides a more systematical and convenient handling of the solution process for the nonlinear equations. Its key point is to reduce the given PDEs to variable-coefficient ordinary differential equations, then we look for solutions to the resulting equations by some methods. As an application, exact solutions for the KdV equation are formally derived.

Approximate homotopy symmetry and infinite series solutions to the perturbed mKdV equation

We devise a homotopy model for the perturbed mKdV equation with weak fourth order dispersion and weak dissipation, and demonstrate that the related approximate equations lead to similarity reduction equations and similarity reduction solutions of different orders which are formally coincident, respectively. Series reduction solutions for the perturbed mKdV equation are thus derived. Painlevé II type equations or hyperbolic secant function solutions are obtained for zero-order similarity reduction equations. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations.

Lamb wave propagation in the functionally graded piezoelectric–piezomagnetic material plate

The propagation behaviour of Lamb waves in the functionally graded piezoelectric–piezomagnetic material plate with material parameters varying continuously along the thickness direction is investigated in this paper. The power series technique is employed to solve these variable coefficient ordinary differential equations. Dispersion equations are given for different boundary conditions. In numerical examples, the influence of the variation of each parameter on dispersion curves and cut-off frequency in electrically and magnetically open cases is discussed in detail. Results show that the elastic parameters and density varying along the thickness direction obviously influence the variation of phase velocity. Some variations in electric and magnetic parameters also affect the phase velocity but the influence is too small, while others almost cannot affect the dispersion curves. Cut-off frequency is closely related to two elastic parameters and to density, whereas other parameters almost cannot influence it. All the results can provide theoretical guidance not only for the analysis and design of a magnetoelectric transducer using functionally graded materials, but also for ultrasonic nondestructive evaluation.

The approximate homotopy symmetry reduction for far-field model equation

The far-field model equation is investigated by the approximate homotopy symmetry method. Homotopy series solutions are constructed through summarizing the relevant general formulas for similarity reduction solutions and similarity reduction equations of differernt orders. Similarity reduction equations of different orders are linear variable coefficients ordinary differential equations, and can be solved one by one from zero-order similarity reduction equations. The auxiliary parameter in the homotopy model affects convergence of homotopy series solutions.

Approximate Similarity Reduction for Perturbed Kaup—Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.

A function transformation method and exact solutions to a generalized sinh-Gordon equation

Based on a transformed Painlevé property and the variable separated ODE method, a function transformation method is proposed to search exact solutions to some partial differential equations (PDEs) with hyperbolic or exponential functions. The new approach provides a more systematical and convenient handling of the solution process for the nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painlevé property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we look for solutions to the resulting equations by some methods. As an application, exact solutions for a generalized sinh-Gordon equation are formally derived.

Exact Solutions to a Combined sinh-cosh-Gordon Equation

Based on a transformed Painlevé property and the variable separated ODE method, a function transformation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painlevé property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we seek for solutions to the resulting equations by some methods. As an application, exact solutions for the combined sinh-cosh-Gordon equation are formally derived.

Approximate similarity reduction for the perturbed mKdV equation via symmetry perturbation and direct method

The perturbed mKdV equation with the general form is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods; series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations. By comparison, the relation between the results obtained by applying the two methods is revealed.