Order Ordinary Differential Equations Research Articles

Nontrivial solutions of fourth order ordinary differential equations with nonlocal three-point boundary conditions

Nontrivial solutions of fourth order ordinary differential equations with nonlocal three-point boundary conditions

Rational solutions to the Painlevé II equation from particular polynomials

The Painlevé equations were derived by Painlevé and Gambier in 1895-1910. Given a rational function R in w, w′ and analytic in z, they searched what were the second order ordinary differential equations of the form w″ = R(z, w, w′) with the properties that the singularities other than poles of any solution or this equation depend on the equation only and not of the constants of integration. They proved that there are 50 equations of this type, and the Painlevé II is one of these. Here, we construct solutions to the Painlevé II equation (PII) from particular polynomials. We obtain rational solutions written as a derivative with respect to the variable x of a logarithm of a quotient of a determinant of order n + 1 by a determinant of order n. We obtain an infinite hierarchy of rational solutions to the PII equation. We give explicitly the expressions of these solutions solution for the first orders.

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G′/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

A new Sixth-Order Runge - Kutta - Nystro ̈m Method for Special Second Order Ordinary Differential Equations

this article, an explicit Runge-Kutta-Nystr m method of sixth order for solving directly second-order ordinary differential equations is constructed. The stability property of the new method is discussed. Numerical illustrations are presented to show the efficiency of the new method by comparing it with other existing Runge-Kutta-Nystr m and Runge-Kutta methods in the scientific literature.

One–Step Hybrid Block Scheme for the Numerical Approximation for Solution of Third Order Initial Value Problems

In this research, we have proposed the simulation of linear block algorithm for modeling third order highly stiff problem without reduction to a system of first order ordinary differential equation, to address the weaknesses in reduction method. The method is derived using the linear block method through interpolation and collocation. The basic properties of the block method were recovered and was found to be consistent, convergent and zero-stability. The new block method is been applied to model third order initial value problems of ordinary differential equations without reducing the equations to their equivalent systems of first order ordinary differential equations. The result obtained on the process on some sampled modeled third order linear problems give better approximation than the existing methods which we compared our result with.

Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast

Abstract In order to improve the modelling efficiency in dynamic system prediction, this paper proposes a predictive model based on high-order normal differential equations to model high-order differential data to obtain an explicit model. The high-order constant differential equation model is reduced, and the numerical method is used to solve the predictive value. The results show that the method realises the synchronisation of model establishment and parameter optimisation, and greatly enhances the modelling efficiency.

Significance of viscosity and thermal conductivity variation parameters on the dynamics of Newtonian fluid conveying tiny particles over a convectively heated surface

The significance of viscosity and thermal conductivity variation parameters on the dynamics of Newtonian fluid conveying tiny particles over a convectively heated surface is examined. The dimensional partial differential equations controlling the flow are remodeled to ordinary differential equations via suitable similarity variables in conjunction with the rescaled Nusselt, Sherwood and density number of motile microorganisms. The resulting nonlinear coupled ordinary differential equations are consequently scaled down to a system of first order ordinary differential equations. The system of equations are evaluated arithmetically via shooting technique along with fourth order Runge–Kutta integration scheme for different boundary conditions. The result was found to be in excellent agreement when compared with available records in the literature. The major highlights of the problem are analyzed and discussed thoroughly. It is seen that increase in viscosity variation parameter is capable to cause a decline in the local skin friction coefficients at the rate of − 0 . 04985 and the Nusselt number is an increasing function of the thermal conductivity variation parameter.

Adaptive Backstepping Method for Stabilizing Systems of Fractional Order Ordinary Differential Equations

In this paper the method of adaptive backstepping for stabilizing and solving system of ordinary and partial differential equations will be used and applied to investigate and study the stability linear systems of Caputo fractional order ordinary differential equations. The basic idea of this approach is to find a quadratic Lyapunov functions for stabilizing the subsystems.

Computational study of a 3-step hybrid integrators for third order ordinary differential equations with shift of three off-step points

A Block of hybrid method with three off-step points based is presented in this work for direct approximation of solution of third-order Initial and Boundary Value Problems (IVPs and BVPs). This off-step points are formulated such that they exist only on a single step at a time. Hence, these points are shifted to three positions respectively in order to obtain three different integrators for computational analysis. These analysis includes; order of the methods, consistency, stability and convergence, global error, number of functions evaluation and CPU time. The superiority of these methods over existing methods is established numerically on different test problems in literature

A New Special 15-Step Block Method for Solving General Fourth Order Ordinary Differential Equations

A new higher-implicit block method for the direct numerical solution of fourth order ordinary differential equation is derived in this research paper. The formulation of the new formula which is 15-step, is achieved through interpolation and collocation techniques. The basic numerical properties of the method such as zero-stability, consistency and A-stability have been examined. Investigation showed that the new method is zero stable, consistent and A-stable, hence convergent. Test examples from recent literature have been used to confirm the accuracy of the new method.

Decoupling of second order systems via linear time invariant transformations

Decoupling of linear time invariant (LTI) systems of second order ordinary differential equations (ODEs) with real matrix coefficients via LTI transformations in the configuration space is revisited. Necessary and sufficient conditions for the existence of such transformations are presented, as well as methods to construct transformation matrices. Matrix series and general matrix function formulas for the system’s matrix coefficients which enable decoupling are also discussed. Systems with non-symmetric and symmetric matrices are considered. Situations when systems degenerate into second order ODEs and lower order ODEs, including differential algebraic equations (DAEs), are analyzed as well.

Asymmetric 3D stress- and flux-induced wave propagation in transversely isotropic thermoelastic solids by using of analytical methods

Asymmetric 3D wave propagation in transversely isotropic thermoelastic solids is presented by using of potential functions and integral transforms. Governing equations of thermoelastodynamic boundary-value problem (TBVP) are including three equations of motion and an energy equation which make a set of four coupled partial differential equations (PDE) in terms of displacements and temperature. By using of the displacement- and temperature-potential function relationships, four coupled equations of TBVP are reduced to two uncoupled PDEs governing the potential functions which are of sixth and second orders. To solve these uncoupled PDEs, complex Fourier expansions and Hankel integral transforms are used in this paper to suppress the angular and radial variables respectively in cylindrical coordinate system. Consequently, sixth and second order ordinary differential equations (ODE) in terms of depth are received. By solving ODEs, satisfying the radiation and boundary conditions and applying the Hankel inverse transforms, the displacement and temperature functions have been determined in the real space. To validate the analytical and numerical results, some comparisons have been made with the results presented in the literature where excellent agreements have been observed. Important points have been mentioned about speed of SH, SV, P and Rayleigh waves in transversely isotropic and isotropic thermoelastic solids.

Numerical Solution of Fourth-Order Boundary Value Problems for Euler-Bernoulli Beam Equation using FDM

Euler-Bernoulli beam equation is widely used in engineering, especially civil and mechanical engineering to determine the deflection or strength of bending beam. In physical science and engineering, to predict the deflection for beam problem, bending moment, soil settlement and modeling of viscoelastic flows, fourth-order ordinary differential equation (ODE) is widely used. The analytical solution of most of the higher order ordinary differential equations with complicated boundary condition that occur in any engineering problems is not easy way. Therefore, numerical technique based on finite difference method (FDM) is comparatively easy and important for solving the boundary value problems (BVP). In this study four boundary conditions (Neumann condition) are considered for solving BVP. Absolute error calculation, numerical stability and convergence are discussed. Two examples are considered to illustrate the finite difference method for solving fourth order BVP. The numerical results are rapidly converged with exact results. The results shows that the FDM is appropriate and reliable for such type of problems. Thus present study will enhance the mathematical understanding of engineering students along with an application in different field.

Mathematical analysis of some couplings for the Eddington-inspired-Born-Infeld theory of bi-gravity in Bondi coordinates

In the Eddington-inspired-Born-Infeld theory, we consider a metric tensor g, an auxiliary tensor q, a scalar field and an electromagnetic field in Bondi coordinates. Using ‘the null tetrad’ associated with each metric, we derive a system of nonlinear partial differential equations and, after some reduction process, we obtain a second order ordinary differential equation coupled to a first order partial differential equations with strong nonlinearities. Using both the solution-tube concept and the nonlinear analysis tools such as the fixed point theorem, we prove an existence and uniqueness result for the nonlinear system obtained.

Linear first order Riemann-Liouville fractional differential and perturbed Abel's integral equations

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's integral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical problems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is different from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives.

Investigation of boundary stresses on MHD flow in a convergent/divergent channel: An analytical and numerical study

This study investigates the boundary stresses prescribed by the traction boundary condition on an incompressible magnetohydrodynamic Jeffery-Hamel flow of viscous fluid. The governing partial differential equations are transformed to a nonlinear ordinary differential equation (ODE) by the transformation obtained from continuity equation. We have modeled traction boundary conditions to solve the problem. We have not imposed flow symmetry condition and applied prescribed flow rate condition. We have computed analytical and numerical solutions of the problem. We have developed modified Adomian decomposition method based on Daun-Rach Approach (DRA) for general nonlinear third order ODE subject to Robin’s and integral boundary conditions. The governing equation is solved using the developed scheme. The Mathematica routine, NDSolve, is used to obtain numerical solution by first approximating the constant flow rate condition by Gauss-Lobatto integration formula of four and five points as well as by trapezoidal rule. Analytical and numerical results are compared and found in good agreement. The slip and no-slip scenarios are observed both for inertial and non-inertial flows. The back flow occurs both for convergent and divergent channel geometries. Flow bifurcations and boundary layer get strengthened due to presence of boundary stresses. The symmetric and asymmetric natures of flow are also captured due to presence of boundary stresses.

Finite difference simulations for magnetically effected swirling flow of Newtonian liquid induced by porous disk with inclusion of thermophoretic particles diffusion

Heat and mass transfer analysis of viscous liquid flow generated by rotation of disk has generated prodigious interest due to promising utilizations in numerous processes such as thermal energy generation systems, turbine rotators, geothermal energy preservations, chemical processing, medicinal instrumentations, computing devices and so forth. In view of such extraordinary utilizations in numerous engineering procedures existent exertion is excogitated to disclose flowing phenomenon over rotating disk. To raise the importance of current analysis influential physical aspects like magnetic field, permeability, Dufour and Soret diffusion phenomenon are also incorporated. Subsequently, flow field distributions are analyzed for suction and injection cases. Modelling is structured via PDE’s by obliging constitutive conservation laws. Boundary layer approach is executed to reduce complexity of attained partial differential system. Transformations developed by Karman are implemented to convert developed differential framework into ODE’s. Implicitly finite differenced technique known as Keller Box is engaged to find solution of coupled intricate high order ordinary differential equations. Influence of flow controlling parameters on associated distributions are revealed through graphical and tabular representations. The related quantities of engineering interest like coefficients of wall drag force, along radial and tangential directions are also computed. Credibility of presently computed results is established by constructing comparison with previously published literature. It is inferred that magnetic strength parameter enhances tangential and radial components of velocity whereas contrary trend is depicted in axially directed velocity. In addition, temperature and momentum distributions show up surging attribute versus magnetic field parameter. All associated profiles have exhibited decrementing aspects against suction parameter. It is also revealed that increment in Soret tends to produce depreciation in temperature profile whereas concentration distribution is enhanced.

WKB periods for higher order ODE and TBA equations

We study the WKB periods for the (r + 1)-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the {A}_r^{(1)} affine Toda field equation. We compute the quantum corrections by using the Picard-Fuchs operators. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra Ar. For the quadratic potential, we propose a formula to show the equivalence between the logarithm of the Y-function and the WKB period, which is confirmed by solving the TBA equation numerically.

Linear connections and shape maps for second order ODEs with and without constraints

We deal with the construction of linear connections associated with second order ordinary differential equations with and without first order constraints. We use a novel method allowing glueing of submodule covariant derivatives to produce new, closed form expressions for the Massa-Pagani connection and our extension of it to the constrained case.

SOLUTION OF THE SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS USING ADOMIAN DECOMPOSITION METHOD

In this paper, the Adomian Decomposition Method (ADM) is employed in solving second order ordinary differential equation. Numerical algorithm was developed. The decomposition method provides a solution as an infinite series in which terms can easily be determined. It is observed that the method is practically suited for initial value problems. The method is effective and easy to implement. The results were presented in both tabular and graphical forms.