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In this paper we gave a general formula for particular solution of the non-homogenous linear ordinary differential equation in a method of variation of parameters of kth order. We solved the linear...

Solutions to nonlinear boundary value problems modelling physical phenomena in engineering applications have traditionally been approximated using numerical methods. More recently, several semi-analytical methods have been developed and used extensively in diverse engineering applications. This work compares the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems arising in engineering. The accuracy and efficiency of the method of variation of parameters are compared to those of two widely used semi-analytical methods, the Adomian decomposition method and the differential transformation method, for three practical engineering boundary value problems: (1) the deflection of a cantilevered beam with a concentrated load, (2) an adiabatic tubular chemical reactor which processes an irreversible exothermal reaction, and (3) the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylinder conduit. The accuracy and convergence of each method is investigated using the error remainder function. The method of variation of parameters is significantly more efficient than both the semi-analytical methods and traditional numerical methods while maintaining comparable accuracy. Unlike the semi-analytical methods, the efficiency of variation of parameters is independent of the nonlinearity. Variation of parameters is shown to be an attractive alternative to semi-analytical methods and traditional numerical methods for solving boundary value problems encountered in engineering applications in which solution efficiency is important.

Lane-Emden type equation is a nonlinear differential equation appears in many fields such as stellar structure, radioactive cooling and modeling of clusters of galaxies. In this work, this equation is investigated using a semi-analytical method called the Variation of parameters method with an auxiliary parameter. In the applied technique, an unknown auxiliary parameter is inserted in Variation of Parameters Method to solve some special cases of these equations. The used algorithm is easy to implement and very effective. The obtained solutions are also fairly accurate.

In the case of nonlinear problems, whether in differential or difference equations, it is difficult and in some cases impossible to invert the problem and obtain a suitable mapping that can be effectively used in fixed point theory to qualitatively analyze its solutions. In this paper we consider the existence of a positive sequence and utilize it in the capacity of integrating factor to obtain a new variation of parameters formula. Then, we will use the obtained new variation of parameters formula and revert to the contraction principle to arrive at results concerning, boundedness, periodicity and stability. The author is working on parallel results for the continuous case. Resumen En el caso de problemas no-lineales, ya sea en ecuaciones diferenciales o en diferencias, es difícil y en algunos casos imposible invertir el problema y obtener una aplicación apropiada que pueda ser efectivamente usada en teoría de punto fijo para analizar quantitativamente sus soluciones. En este paper consideramos la existencia de una sucesión positiva y la usamos en la capacidad del factor de integración para obtener una nueva fórmula de variación de parámetros. Luego, usaremos esta nueva fórmula de variación de parámetros y volver al principio de contracción para obtener resultados que involucran acotamiento, periodicidad y estabilidad. El autor se encuentra trabajando en resultados paralelos para el caso continuo

This work presents new approximate analytical solutions for the Riccati equation (RE) resulting from the application of the method of variation of parameters. The original equation is solved using another RE explicitly dependent on the independent variable. The solutions obtained are easy to implement and highly applicable, which is confirmed by solving several examples corresponding to REs whose solution is known, as well as optimizing the method to determine the density of the members that make up a population. In this way, new perspectives are open in the study of the phenomenon of pattern formation.

SummaryIn the method of variation of parameters, a sufficient condition is often imposed in order to obtain a particular solution for a higher order linear nonhomogeneous differential equation. We ...

This work considers LFIVP defined under gH-differentiability with the aim of establishing a method and subsequently, a fuzzy-valued function, which is a solution to the LFIVP. The conditions for a fuzzy function to be H- differentiable and gH-differentiability are defined. It is however established that gH-differentiability concept to fuzzy-valued functions is more generalized than H-differentiable. Characterization and Stacking theorems are also used to translate LFIVP to a system of specialized ODEs. Method of variation of parameters is also extended to LFIVP. Examples are constructed to test the applicability of the methods and graphical presentation of the behavior of solutions of the constructed examples is done with the aid of Matlab 8.5 version. We recommended that further studies should be made on second and higher order LFIVP.

In this article flow of laminar, isothermal, incompressible viscous fluid is considered in a rectangular domain with infinite length and bounded by two moving porous walls which enable the fluid to enter or exit during successive contractions or expansions. Solution to the problem is approximated by using variation of parameters method (VPM). To investigate the effect of non-dimensional wall expansion/contraction rate $$\alpha $$ and permeation Reynolds number R on the flow field, graphical results are presented. Analytical solution obtained by (VPM) is supported by numerical results andboth show excellent agreement. The study of the flow between dilating or squeezing porous walls is drastic simplification of the transport of biological fluids through dilating or squeezing vessels.

Nonlinear problems arise in many heat transfer applications, and several analytical and numerical methods for solving these problems are described in the literature. Here, the method of variation of parameters is shown to be a relatively simple method for obtaining solutions to four specific heat transfer problems: 1. a radiating annular fin, 2. conduction-radiation in a plane-parallel medium, 3. convective and radiative exchange between the surface of a continuously moving strip and its surroundings, and 4. convection from a fin with temperature-dependent thermal conductivity and variable cross-sectional area. The results for each of these examples are compared to those obtained using other analytical and numerical methods. The accuracy of the method is limited only by the accuracy with which the numerical integration is performed. The method of variation of parameters is less complex and relatively easy to implement compared to other analytical methods and some numerical methods. It is slightly more computationally expensive than traditional numerical approaches. The method presented may be used to verify numerical solutions to nonlinear heat transfer problems.

This article is devoted to establish a novel analytical solution scheme for the fractional diffusion equations. Caputo’s formulation followed by the variation of parameters method has been employed to obtain the analytical solutions. Following the derived analytical scheme, solution of the fractional diffusion equation for several initial functions has been obtained. Graphs are plotted to see the physical behavior of obtained solutions.

The method of variation of parameters still has a great interest and wide applications in mathematics, physics and astrodynamics. In this paper, universal functions (the Y's functions) based on Goodyear's time transformation formula were used to establish a variation of parameters method which is useful in slightly perturbed two-body initial value problem. Moreover due to its universality, the method avoids the switching among different conic orbits which are commonly occurring in space missions. The position and velocity vectors are written in terms of f and g series. The method is developed analytically and computationally. For the analytical developments, exact literal formulations for the differential system of variation of the epoch state vector are established. Symbolical series solution of the universal Kepler's equation was also established, and the literal analytical expressions of the coefficients of the series are listed in Horner form for efficient and stable evaluation. For computational developments of the method, an efficient algorithm was given using continued fraction theory. Finally, a short note on the method of solution was given just for the reader guidance.

The method of variation of parameters can be found in most undergraduate textbooks on differential equations. The method leads to solutions of the non-homogeneous equation of the form , a sum of function products using solutions to the homogeneous equation and . Why though, might varying parameters of solutions to the homogeneous equation be a feasible approach? An insight is needed, even for a good guess. This paper is for instructors teaching a first course in undergraduate differential questions, where the students already have covered first-order linear equations and second-order homogeneous linear equations. We provide a variation to the standard textbook approach to variation of parameters, intended to help the student find their way toward a key insight: “Aha! Let's try …” We point to several reasons why it can be important for students to learn variation of parameters.

This paper establishes variation of parameters formula for impulsive differential equations with initial time difference. As an application, one of the results is used to investigate stability properties of solutions.

It is well known that nonlinear integro-differential equations play vital role in modeling of many physical processes, such as nano-hydrodynamics, drop wise condensation, oceanography, earthquake and wind ripple in desert. Inspired and motivated by these facts, we use the variation of parameters method for solving system of nonlinear Volterra integro-differential equations. The proposed technique is applied without any discretization, perturbation, transformation, restrictive assumptions and is free from Adomian’s polynomials. Several examples are given to verify the reliability and efficiency of the proposed technique.

This paper proposes a novel approach to include stochastic variation in the framework of a macroscopic traffic model. Simultaneously, the effects of insufficient data quality and consequent inaccurate identification are investigated. The traffic density distribution over time and space provides a deeper understanding of the evolution of jams, which in turn is vital for the development of traffic network control algorithms. Model parameters are estimated using both Eulerian and Lagrangian sensing. Instead of using deterministic model parameters directly, this work proposes to use stochastic model parameters. Therefore, in addition to the estimated mean of model parameters, a corresponding covariance matrix is estimated as well. The utilized traffic model supports arbitrary Fundamental Diagrams defined by (a combination of) model parameters. Using an analytic approach to define the resulting variation in the Fundamental Diagram is generally impossible, due to the nonlinear combination of random vari...

Starting from the solutions of soliton equations and corresponding eigenfunctions obtained by Darboux transformation, we present a new method to solve soliton equations with self-consistent sources (SESCS) based on method of variation of parameters. The KdV equation with self-consistent sources (KdVSCS) is used as a model to illustrate this new method. In addition, we apply this method to construct some new solutions of the derivative nonlinear Schrodinger equation with self-consistent sources (DNLSSCS) such as phase solution, dark soliton solution, bright soliton solution and breather-type solution.

In this paper, we develop a reliable algorithm which is called the variation of parameters method for solving sixth-order boundary value problems. The proposed technique is quite ecient and is practically well suited for use in these problems. The suggested iterative scheme finds the solution without any perturbation, discritization, linearization or restric- tive assumptions. Moreover, the method is free from the identification of Lagrange multipliers. The fact that the proposed technique solves nonlin- ear problems without using the Adomian's polynomials can be considered as a clear advantage of this technique over the decomposition method. Several examples are given to verify the reliability and eciency of the proposed method. Comparisons are made to reconfirm the eciency and accuracy of the suggested technique.

In this paper we obtain variation of parameters formula for differential equations with causal maps. Mathematics subject classification (2000): 34D20.

It is known that Alekseev's variation of parameters formula for ordinary differential equations can be generalized to other types of causal equations (including delay differential equations and Volterra integral equations), and corresponding discrete forms. Such variation of parameters formulae can be employed, together with appropriate inequalities, in discussing the behaviour of solutions of continuous and discretized problems, the significance of parameters in mathematical models, sensitivity and stability issues, bifurcation, and (in classical numerical analysis) error control, convergence and super-convergence of densely defined approximations and error analysis in general. However, attempts to extend Alekseev's formula to nonlinear Volterra integral equations are not straightforward, and difficulties can recur in attempts to analyze the sensitivity of functionally-dependent or structurally-dependent solutions. In analyzing sensitivity we discuss behaviour for infinitesimally small perturbations. In discussions of stability we need to establish the existence of bounds on changes to solutions (or their decay in the limit as t->~) that ensue from perturbations in the problem. Yet the two topics are related, not least through variation of parameters formulae, and (motivated by some of our recent results) we discuss this and related issues within a general framework.

Given a basis of solutions to k ordinary linear differential equations lj[y]=0 (j=1,2,…,k), we show how Green's functions can be used to construct a basis of solutions to the homogeneous differential equation l[y]=0, where l is the composite product l=l1l2…lk. The construction of these solutions is elementary and classical. In particular, we consider the special case when l=l1k. Remarkably, in this case, if {y1,y2,…,yn} is a basis of l1[y]=0, then our method produces a basis of l1k[y]=0 for any k∈N. We illustrate our results with several classical differential equations and their special function solutions.

In this chapter we stress the importance of nonlinear variation of parameters formulae for a variety of nonlinear problems as well as monotone iterative technique that offers constructive methods for the existence of solution in a sector.

We define the principal matrix solution $Z(t,s)$ of the linear Volterra vector integro-differential equation \[ x'(t) = A(t)x(t) + \int_s^t B(t,u)x(u)\,du \] in the same way that it is defined for $x' = A(t)x$ and prove that it is the unique matrix solution of \[ \frac{\partial}{\partial{t}}Z(t,s) = A(t)Z(t,s) + \int_{s}^t B(t,u)Z(u,s)\,du, \quad Z(s,s) = I. \] Furthermore, we prove that the solution of \[ x'(t) = A(t)x(t) + \int_{\tau}^t B(t,u)x(u)\,du + f(t), \quad x(\tau) = x_0\] is unique and given by the variation of parameters formula \[ x(t) = Z(t,\tau)x_0 + \int_{\tau}^t Z(t,s)f(s)\,ds.\] We also define the principal matrix solution $R(t,s)$ of the adjoint equation \[ r'(s) = -r(s)A(s) - \int_s^t r(u)B(u,s)\,du \] and prove that it is identical to Grossman and Miller's resolvent, which is the unique matrix solution of \[ \frac{\partial}{\partial{s}}R(t,s) = -R(t,s)A(s) - \int_{s}^t R(t,u)B(u,s)\,du, \quad R(t,t) = I. \] Finally, we prove that despite the difference in their definitions $R(t,s)$ and $Z(t,s)$ are in fact identical.

Nonlinear variation of parameters formula for set differential equations in a metric space is developed investigating necessary tools such as differentiability of solutions relative to initial values. Other needed results are also proven.

We show that a method of embedding for a class of non-linear Volterra equations can be used in a novel fashion to obtain variation of parameters formulas for Volterra integral equations subjected to a general type of variation of the equation. The approach is of intrinsic interest. Our variation of parameters formulas generalize classical formulas for ordinary differential equations (due to Alekseev) and for linear Volterra integral equations (based on resolvents). Illustrative examples are related to known results.

This article investigates the relationship between an unperturbed integrodifferential system and a perturbed integrodifferential system that have different initial positions and an initial time difference (ITD). Variation of parameter techniques are employed to obtain integral formulae and to define Lyapunov-like functions.

We have discovered that application of the method of variation of parameters to the KustaanheimoStiefel (K-S) regularization drastically reduces the orbital integration errors of the perturbed two-body problem for arbitrary types of perturbations. This is because not only the errors of position, whose linear growth was determined previously (Paper I), but those of the physical time grow only linearly with respect to the —ctitious time even if using traditional integrators such as the Runge-Kutta, extrapolation

A variation of parameters formula and Gronwall type integral inequality are proved for a differential equation involving general piecewise alternately advanced and retarded argument.

We exhibit classes of scalar and vector linear differential equations with functional derivatives for which the Lagrange method of variation of parameters can be applied.

Here is a discovery‐type proof of a strong theorem on variation of parameters. The theorem avoids any continuity assumptions on the given functions; and the proof uses little more than the definition of the Riemann integral.

A variation of parameters and Gronwall type integral inequality are proved for a differential equation involving piecewise alternately retarded and advanced argument.

This paper provides a complete, direct derivation of the continuous time linear system H/sub /spl infin// filter using a variation of parameters approach. Conditions, derived from the variational technique, which provide for filter existence, performance, and stability, are presented. These results are consistent with existing derivations using similar techniques. Here, the results also provide a simple relationship between the matrix Riccati equation for the filter gain calculation and the matrix Riccati equation for the bounded real lemma.

This paper focuses on studying the integration method of a generalized Birkhoffian system. The method of variation on parameters for the dynamical equations of a generalized Birkhoffian system is presented. The procedure for solving the problem can be divided into two steps: the first step, a system of auxiliary equations is constructed and its general solution is given; the second step, the parameters are varied, and the solution of the problem is obtained by using the properties of generalized canonical transformation. The method of variation on parameters for the generalized Birkhoffian system is of universal significance, and we take a nonholonomic system and a nonconservative system as examples to illustrate the application of the results of this paper.

In this paper, we use the variation of parameters method to solve a class of eighth-order boundary-value problems. The analytical results are calculated in terms of convergent series. Error estimates are tabulated and are compared with some other well known techniques which reveal the complete reliability of the suggested scheme. Several examples are given to verify the reliability and efficiency of the proposed method.

The method of variation of parameters invented by Lagrange (1, 2) in 1774 remains the most powerful procedure for obtaining particular solutions of nonhomogeneous ordinary linear differential equations. The general outlines of the method of variation of parameters also have been utilized, in an astonishingly successful manner, to obtain the approximate solution of certain nonlinear ordinary differential equations, in a method associated with the names of Kryloff and Bogoliuboff (3). A feature of both methods which is not motivated nor made plausible by these and other writers, in particular of textbooks, is the apparently arbitrary artifice of requiring that a certain linear combination (of solutions of the null differential equation corresponding to the nonhomogeneous equation) be made to vanish. Here it is shown that when the linear differential equation is of the second order, 1 and the linear combination concerned is required to be a differentiable function f rather than zero (in the particular solution obtained using this condition), the terms containing f vanish identically and the solution reduces to that given by the classical method of variation of parameters. Thus Lagrange's choice of zero for f, rather than being arbitrary, may be justified on the ground that it is the most judicious one for a function whose form does not matter and which therefore may be determined on the sole basis of convenience. In contrast to the just-noted situation with the method of variation of parameters, it is shown here that the previously-noted condition on the linear combination in question is essential to the method of Kryloff and Bogoliuboff. That is, if this linear combination is required to be an analytic function F, the Kryloff-Bogoliuboff “equations of the first approximation” are not recovered unless F is identically zero.

In this paper, the transient response of a third-order non-linear system is obtained by first reducing the given third-order equation to three first-order equations by applying the method of variation of parameters. On the assumption that the variations of amplitude and phase are small, the functions are expanded in ultraspherical polynomials. The expansion is restricted to the constant term. The resulting equations are solved to obtain the response of the given third-order system. A numerical example is considered to illustrate the method. The results show that the agreement between the approximate and digital solution is good thus vindicating the approximation.

A new formulation is presented for the perturbed Lambert problem. The formulation employs the variation-of-parameters method in the KS transformed state space to determine perturbations of a Keplerian Lambert solution. The approach is universal (in that its validity is not restricted to a particular energy domain). For the case of the second zonal harmonic (oblateness) perturbation, first order perturbations are carried out entirely analytically; non-iterative corrections are determined through solution of a pair of algebraic equations. For more general perturbations, numerical quadratures are required.

A nonlinear variation of parameters method for ordinary differential equations is obtained, which generalizes existing techniques. The versatility of the method is demonstrated by studying the asymptotic behavior of solutions of perturbed equations.

In the ordinary method of variation of parameters, a solution of a given linear differential equation is assumed to be a linear combination of independent solutions of an auxiliary solvable equation, and the coefficients in that linear combination are functions to be determined. A new version presented in this paper uses a linear combination including also the first derivative of the solution of an auxiliary equation. We have applied this method successfully to the stepwise numerical integration of the partialwave Schrodinger equation in scattering theory. Since higher-order terms in a powerseries solution can be included rather straightforwardly, the new method achieves high accuracy even when the step size is appreciably large. A numerical example is presented.

The variation of parameters formula is an important tool for the study of both ordinary and retarded differential equations. Recently, a nonlinear extension of this formula, due to Alekseev, has proved useful in studying perturbations of nonlinear ordinary differential equations. (See [1], [2], [3], and [7].) It is the purpose of this note to show that this formula may also be extended to retarded equations, thus generalizing the result given, for example, in [5, p. 366] for the linear case. This allows one to extend to equations with finite lag many of those results in [2], [3], and [7] which assume a bound on the solutions or their "derivatives with respect to initial conditions." We first discuss the differential-difference equation

Recent work in stability theory for linear ordinary differential equations has made much use of the inequalities which can be deduced from the variation-of-parameters formula. In this article it is shown that similar inequalities hold for nonlinear Stieltjes integral equations and hence, as a special case, for nonlinear differential equations in which the nonlinearity is of Lipschitz type.

In this paper we present a nonlinear variation of parameters method for a neutral functional differential equation, without the help of the associated variational equation. An application is made to the study of the asymtotic behavior of solutions of perturbed equations.

On developpe une variation non lineaire de la formule des parametres pour des equations integrodifferentielles. On etudie le probleme de la continuite et de la differentiabilite des solutions des equations integrodifferentielles non lineaires et on obtient la relation entre les derivees

A new 3D chaotic system with only one nonlinear term is constructed in this paper, which has conservative and dissipative features respectively by choosing different parameters. The complex dynamic behaviors of the system are analyzed theoretically and numerically by using nonlinear analytical tools, such as Lyapunov exponents, phase portraits, and bifurcation diagrams. In a certain range of parameters, the new system is a non-Hamiltonian conservative chaotic system, which can produce infinitely many coexisting conservative flows. With the variation of parameters, the divergence of the system is less than zero, and the system can also generate hidden and self-excited attractors with multi-scroll. To verify the feasibility of the new system, it has been implemented by field-programmable gate array; phase diagrams obtained in the experiment are consistent with the simulation of the data.

This study proposed an efficient numerical technique for coupled differential equations (CDEs) using the clique polynomials of the Complete graph. Recently, Graph theory has dragged the attention of many mathematicians due to its wide applications. Here, some problems have been considered to examine the proposed scheme proficiency. Some theorems on convergence are discussed. Here, the CDEs are rehabilitated into an algebraic equation system using the clique polynomials and collocation technique. The proposed scheme results are compared with the undetermined coefficients (UDCM) and variation of parameters method (VPM) solutions through tables and graphs. Obtained results reveal that the current approach is more accurate.

A direct method to optimize parameters in nonorthogonal basis orbitals is discussed. The partial derivatives of the energy of the state of interest, not necessarily the ground state, with respect to the orbital parameters are calculated analytically. The required cofactors up to third order of the overlap matrices over spin-space orbitals in pairs of Slater determinants are calculated by biorthogonalization. All parameters are varied simultaneiously in the direction of the negative gradient (steepest descent). A search logic with dynamically adjusted step-size is shown, in the example of the minimum basis Ne ground state, to find the energy minimum in an efficient manner.

In this paper, we use the modified variation of parameters method (MVPM), an elegant coupling of variation of parameters method (VPM) and Adomian's decomposition method (ADM), for finding the analytical solution of system of nonlinear fractional boundary value problems associated with obstacle. Caputo sense of fractional derivative is used to coup up with fractional term. The results are calculated in terms of series with easily computable components. The used technique is quite easy and convenient for such type problems because it has been previously applied over several nonlinear obstacle systems.