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In this research paper, we discuss systems of conformable linear differential equations. The conformable fundamental exponential matrix has been used to express the solution of the homogeneous and nonhomogeneous systems. The method of variation of parameters has been investigated to find the particular solution of the conformable system of nonhomogeneous linear differential equations. Several illustrative examples have been provided at the end of our study to validate all obtained results.

In this paper, we study a fourth order linear differential equation. We found an upper bound for the solutions of this differential equation and also, we prove that all the solutions are in L4(0, ∞). By comparing these results we obtain that all the eigenfunction of the boundary value problem generated by this differential equation are bounded and in L4(0, ∞).

In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

A simple, yet accurate modified multi-scale method (MMSM) for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed. This method depends on the classical multi-scale method (MSM) and the method of variation of parameters. Assuming that the forced excitation is a constant, one could easily obtain the approximate analytical solution of the simplified system based on the traditional MSM. Then, this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation. To certify the correctness and precision of the proposed analytical method, the van der Pol system with two scales subject to slowly periodic excitation is investigated; this system presents rich dynamical phenomena such as spiking (SP), spiking-quiescence (SP-QS), and quiescence (QS) responses. The approximate analytical expressions of the three types of responses are given by the MMSM, and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method (HBM). The results obtained by the present method are considerably better than those obtained by traditional methods, quantitatively and qualitatively, particularly when the excitation frequency is far less than the natural frequency of the system.

Supercapacitor (SC) is one of the most trending energy storage solutions. The SCs equivalent circuit models have been extensively applied to energy management because of their accuracy and simplicity. However, there is a difficulty in solving the conventional differential equation-based model used to characterize the electrical behavior of SCs operating in constant power applications. Hence, numerical or metaheuristic techniques have been used in the literature to derive SC's internal voltage at any time. In this work, a thorough mathematical analysis that enables a precise calculation of the electrical variables implied in the charge/discharge processes of SCs operated at constant power as a function of time is presented. First, the transcendental discharge voltage expression of SCs operating at constant power is formulated, and then it was solved using the Special Trans Function theory (STFT). The precision of calculation of the method used for solving the transcendental expression is presented and discussed. Second, the transcendental voltage of charging expression of SCs operating at constant power as a function of time is also formulated, and then it was solved using Lambert W equation. Third, a comparison of different methods for solving mentioned equations is presented. Fourth, the electrical variables involved in the charge/discharge processes of SCs – voltage, current, power, energy, state of charge, and power loss are investigated. Furthermore, the results obtained for the variation of parameters and their operating conditions demonstrate the proposed equations’ applicability and accuracy. Finally, the results obtained validate that the closed-form expressions suggested in this paper are accurate and straightforward, which can contribute to proper modeling, investigation, sizing, regulation, and control of constant power SCs in modern energy systems.

The implementation of the variation of parameters method has been demonstrated for the flow of a Casson fluid through squeezed parallel plates. Governing PDE has been reduced, with the help of similarity transform, to relatively simpler ODE. The consequent non-linear equation is complicated enough to have an exact solution. We have solved that with the help of variation of parameters method. The results are displayed with the help of graphs and are decorated with suitable physical explanation.

We present a geometric proof of the Poincaré-Dulac Normalization Theorem for analytic vector fields with singularities of Poincaré type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field. A similar construction is considered in the case of linearization of maps in a neighborhood of a hyperbolic fixed point.

We present a new method for computing orbits in the perturbed two-body problem: the position and velocity vectors of the propagated object in Cartesian coordinates are replaced by eight orbital elements, i.e., constants of the unperturbed motion. The proposed elements are uniformly valid for any value of the total energy. Their definition stems from the idea of applying Sundman's time transformation in the framework of the projective decomposition of motion, which is the starting point of the Burdet-Ferr\'andiz linearisation, combined with the Stumpff functions. In analogy with Deprit's ideal elements, the formulation relies on a special reference frame that evolves slowly under the action of external perturbations. We call it the intermediate frame, hence the name of the elements. Two of them are related to the radial motion, the next four, given by Euler parameters, fix the orientation of the intermediate frame. The total energy and a time element complete the state vector. All the necessary formulae for extending the method to orbit determination and uncertainty propagation are provided. For example, the partial derivatives of the position and velocity with respect to the intermediate elements are obtained explicitly together with the inverse partial derivatives. Numerical tests are included to assess the performance of the proposed special perturbation method when propagating the orbit of comets C/2003~T4 (LINEAR) and C/1985~K1 (Machholz).

This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay &tau, &gt, 0 , by adding a random forcing term f ( t ) that varies with time: x &prime, ( t ) = a x ( t ) + b x ( t &minus, &tau, ) + f ( t ) , t &ge, 0 , with initial condition x ( t ) = g ( t ) , &minus, &le, t &le, 0 . The coefficients a and b are assumed to be random variables, while the forcing term f ( t ) and the initial condition g ( t ) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L p -solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz&rsquo, s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay &tau, tends to 0, the random delay equation tends in L p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.

With consideration of pre-axial pressure and two-parameter elastic foundation (Pasternak), a new method was put forward for analysis of transverse free vibration of a finite-length Euler-Bernoulli beam resting on a variable Pasternak elastic foundation. Matrices and determinants corresponding to arbitrary boundary conditions were provided for engineers and researchers according to their own demand. In derivation process of new proposed method, Schwarz distribution was adopted for simplifying the partial derivative of Dirac function and compound trapezoidal integral formula was adopted for discretizing the shape function of beam. For the symmetrical boundary conditions, it was concluded that natural frequency of transverse free vibration obtained by FEM highly agreed with the new proposed method. In contrary, for the asymmetrical cases, the calculation results were different from each other. For solving the ordinary differential equation with nonlinear partial derivative terms of shape function, the key point of new proposed method was to establish stiffness equation set composed of obtained matrices, rather than a single equation on the basis of classical theory. This point should be treated as a great advantage. New proposed method can be generalized to solve more complicated problems, which were illustrated in conclusion and prospect.

The existence and smoothness of centre manifolds and a reduction principle are proven for impulsive delay differential equations. Several intermediate results of theoretical interest are developed, including a variation of constants formula for linear equations in the phase space of right-continuous regulated functions, linear variational equation and smoothness of the nonautonomous process, and a Floquet theorem for periodic systems. Three examples are provided to illustrate the results.

This paper gives a representation of a solution to the Cauchy problem for a fractional linear system with pure delay. We introduce the fractional delayed matrices cosine and sine of a polynomial of degree and establish some properties. Then, we use the variation of constants method to obtain the solution and our results extend those for second order linear system with pure delay. As an application, the representation of a solution is used to obtain a finite time stability result.

In this paper, a ( 2 + 1 ) -dimensional Date–Jimbo–Kashiwara–Miwa equation is investigated. Based on the Hirota method and auxiliary variable, Backlund transformation is obtained. Under certain conditions, the Hirota bilinear forms can be reduced to the Plucker and Jacobi identities via the Wronskian and Grammian solutions. Through the Wronskian technique and Pfaffian derivative formulae, N -soliton solutions in the Wronskian and Grammian are given. Graphically presented, soliton collisions are elastic both in the Wronskian and Grammian, and after each collision, the solitons keep their amplitudes unchanged except for the phase shifts.

Several new criteria on exponential stability of a class of impulsive stochastic delay partial differential equations with a fractional Brownian motion are presented by employing the variation of parameters formula and using an inequality technique. The criteria we provided in this paper do not assume the condition that the corresponding continuous stochastic system is stable, so this criteria can be applied to stabilize unstable systems. Moreover, the exponential convergence rate is estimated. Some examples are given to illustrate our main results.

In this paper, we firstly introduce a concept of delayed Mittag-Leffler type matrix function, an extension of Mittag-Leffler matrix function for linear fractional ODEs, which help us to seek explicit formula of solutions to fractional delay differential equations by using the variation of constants method. Secondly, we present the finite time stability results by virtue of delayed Mittag-Leffler type matrix. Finally, an example is given to illustrate our theoretical results.

In this letter, we investigate the ( 2 + 1 ) -dimensional dispersive long wave system, which describes the hydrodynamics of wide channels with finite depth. By using Sato theory and Hirota’s bilinear method, both the Grammian and Wronskian type determinant solution of the bilinear dispersive long wave system are presented. The Wronskian solution to the classical Boussinesq–Burgers system is obtained via direct dimensional reduction.

Applied energy principle analysis natural frequency equations of Euler beam and Timoshenko respectively. The expression of natural frequency is calculated by the separation of variables. Consider the natural frequency ratio between the Euler beam and Timoshenko one to analyze the relationship between geometric parameters and frequency ratio that effect by parameters of frequency. Calculations show that the relationship between frequency ratio and geometric parameters present non-linear parabola characteristic. When the increment of geometric parameters is identical, the frequency value of the beam was lowered by both transverse shear effect and rotary inertia whereby non-linear characteristics are mainly determined by the transverse shear effect.

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.

We model the dynamical behavior of a three dimensional (3-D) dissipative oscillator consisting of a $m$-block whose vertical fall occurs against a spring and which can also slide horizontally on a rigid truss rotating at a known angular speed law $\omega(t)$. The $z$-vertical time law is obvious, whilst its $x$-motion along the horizontal arm is ruled by a linear differential equation to be solved through the Hermite functions and the Confluent Hypergeometric Function (CHF) $_{1}F_{1}$ (Kummer). After the rotation time law $\theta(t)$ has been computed, we know completely the mass motion in a cylindrical coordinate reference: some transients have then been discussed. Finally, further effects as an inclined slide and a contact dry friction have been added to the problem, so that the motion differential equation becomes inhomogeneous and we resort to Lagrange method of variation of constants, helped by a Fourier-Bessel expansion, in order to manage the relevant intractable integrations., Comment: 13 pages, 9 figures. Revised version after some referee's comments

Dynamically varying system parameters along a path enclosing an exceptional point is known to lead to chiral mode conversion. But is it necessary to include this non-Hermitian degeneracy inside the contour for this process to take place? We show that a slow enough variation of parameters, even away from the system's exceptional point, can also lead to a robust asymmetric state exchange. To study this process, we consider a prototypical two-level non-Hermitian Hamiltonian with a constant coupling between elements. Closed form solutions are obtained when the amplification/attenuation coefficients in this arrangement are varied in conjunction with the resonance detuning along a circular contour. Using asymptotic expansions, this input-independent mode conversion is theoretically proven to take place irrespective of whether the exceptional point is enclosed or not upon encirclement. Our results significantly broaden the range of parameter space required for the experimental realization of such chiral mode conversion processes., 4 pages, accepted for publication

Alexseev``s formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In this paper, we investigate bounds for solutions of the functional nonlinear perturbed di®erential systems using the two notion of h-stability and t1- similarity.

The coupled inhomogeneous nonlinear Schrodinger (NLS) equations, which describe the propagation of two nonlinear waves in the inhomogeneous plasma, are investigated. By virtue of the triple Wronskian identities, the coupled inhomogeneous NLS equations are proved to possess the triple Wronskian vector solutions based on the non-isospectral Ablowitz–Kaup–Newell–Segur system. Solving the zero potential Lax pair, we give the bright N-soliton solutions from the triple Wronskian solutions. Amplitude and velocity of the soliton are related to the damping in the plasma. Overtaking interaction, head-on interaction and bound state of the two solitons are given. Solving the non-zero potential Lax pair, we construct the multi-parametric vector rogue-wave solutions of the coupled inhomogeneous NLS equations with the Darboux transformation. Influence of the linear and parabolic density profiles on the background density and amplitude of the rogue wave, is discussed. Bright-dark solitons together with a rogue wave are presented.

This paper is devoted to the study of a diffusive and vector-bias malaria model. We first analyze the well-posedness of the initial value problem of the model. Then, according to the basic reproduction ratio $\mathcal{R}_0$, we establish the existence and non-existence of traveling wave solutions for the model. The proof of the main theorems is based on Schauder fixed point theorem and the variation of constants formula of ODEs.

N-soliton solutions and the bilinear form of the (2 + 1)-dimensional AKNS equation are obtained by using the Hirota method. Moreover, the double Wronskian solution and generalized double Wronskian solution are constructed through the Wronskian technique. Furthermore, rational solutions, Matveev solutions and complexitons of the (2 + 1)-dimensional AKNS equation are given through a matrix method for constructing double Wronskian entries. The three solutions are new.

In this paper, the (3+1)-dimensional BKP equation is mainly being discussed. Based on the the Wronskian technique and the Pfaan properties, Wronskian and Grammian solutions have been established. Generating functions for matrix entries satisfy linear system of partial dierential equations involving a free parameter.

A $(2+1)$ -dimensional nonlinear Schrodinger equation is mainly discussed. Based on the Hirota direct method and the Wronskian technique, multiple-soliton solutions and a generalized double Wronskian determinant are obtained, respectively.

We give a representation of the component of solutions withcharacteristic multiplier 1 in a periodic linear inhomogeneous continuoussystem. It follows from this representation that asymptotic behaviors ofthe component of solutions to the system and to its associated homo-geneous system are quite different, though they are similar in the casewhere the characteristic multiplier is not 1. Moreover, the representationis applicable to linear discrete systems with constant coefficients. 1. Introduction and preliminariesWe consider periodic linear inhomogeneous differential equations of the formddt(1) x(t) = A(t)x(t) +f(t), x(0) = w,where A(t) is a periodic continuous p×p matrix function with period τ > 0 andf : R→ C p a τ-periodic continuous function. Here Cis the set of all complexnumbers and Rthe set of all real numbers.Representations of solutions to the equation (1) have been given in [2, 5, 6].These reformulate the variation of constants formula into the sum of a τ-periodic function and an exponential-like function. In particular, the repre-sentation [6] of the component of solutions with characteristic multiplier µ 6= 1shows that asymptotic behavior of the component of solutions to the equation(1) is the sum of asymptotic behavior of the component of some solution to thehomogeneous equation associated with the equation (1) and the τ-periodic so-lution decided from the equation (1). Roughly speaking, asymptotic behaviorof the component of solutions to the system and to the homogeneous systemassociated with the equation (1) are similar (refer to Remark 3.4). However,it is predicted that in the case where the characteristic multiplier is 1, they

We consider the methods for matrix correction or correction of all parameters of systems of linear equations and inequalities. We show that the problem of matrix correction of an inconsistent system of linear inequalities with the nonnegativity condition is reduced to a linear programming problem. Some stability measure is defined for a given solution to a system of linear inequalities as the minimal possible variation of parameters under which this solution does not satisfy the system. The problem of finding the most stable solution to the system is considered. The results are applied to constructing an optimal separating hyperplane in the feature space that is the most stable to the changes of features of the objects.

Squeezing flow of a viscous fluid is considered. Two types of flows are discussed namely, the axisymmetric flow and two dimensional flow. Similarity transform proposed by Wang (1976) [13] has been used to reduce the Navier–Stokes equations to a highly non-linear ordinary differential equation which jointly describes both types of flows. Solution to aforementioned ordinary differential equation is obtained by using Variation of Parameters Method (VPM). VPM is free from the existence of small or large parameters and hence it can be applied to a large variety of problems as compared to the perturbation method applied by Wang (1976) [13]. Comparison among present and already existing solutions is also provided to show the efficiency of VPM. A convergence analysis is also carried out. Effects of different physical parameters on the flow field is discussed and demonstrated graphically with comprehensive discussions and explanations.

Magneto hydrodynamic (MHD) squeezing flow of a viscous fluid has been discussed. Conservation laws combined with similarity transformations have been used to formulate the flow mathematically that leads to a highly nonlinear ordinary differential equation. Analytical solution to the resulting differential equation is determined by employing Variation of Parameters Method (VPM). Runge–Kutta order-4 method is also used to solve the same problem for the sake of comparison. It is found that solution using VPM reduces the computational work yet maintains a very high level of accuracy. The influence of different parameters is also discussed and demonstrated graphically.

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.

Without the usual assumption of monotonicity, we establish some results on the theory of hyperbolic differential inequalities which enable us to produce a majorising interval function for the solution of the hyperbolic initial value problem. Using this function, a variation of parameters formula and interval iterative technique, the existence of solution to the problem is established.

We consider the system of Volterra integro-dynamic equations and obtain necessary and sufficient conditions for the uniform stability of the zero solution employing the resolvent equation coupled with the variation of parameters formula. The resolvent equation that we use for the study of stability will have to be developed since it is unknown for time scales. At the end of the paper, we furnish an example in which we deploy an appropriate Lyapunov functional. In addition to generalization, the results of this paper provides improvements for its counterparts in integro-differential and integro-difference equations which are the most important particular cases of our equation.

Alexseev`s formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In recent years M. Pinto introduced the notion of -stability. S.K. Choi et al. investigated -stability for the nonlinear differential systems using the notion of -similarity. Applying these two notions, we study bounds for solutions of the perturbed differential systems.

We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a, k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C 0-semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.

This chapter treats the theory of ordinary differential equations, both linear and nonlinear. Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vector-valued $F$ satisfying a “Lipschitz condition,” that the first-order system $y'=F(t,y)$ has a unique local solution satisfying an initial condition $y(t_0)=y_0$. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for $n^{th}$-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in $F$. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves. Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to $n^{th}$-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study—Legendre polynomials and Bessel functions.

We derive Taylor’s theorem using a variation of constants formula for conformable fractional derivatives. This is then employed to extend some recent and classical integral inequalities to the conformable fractional calculus, including the inequalities of Steffensen, Chebyshev, Hermite–Hadamard, Ostrowski, and Gruss.

In this work, a simple derivation of the variable amplitude method using the variation of parameters to solve a differential equation is presented. The variable amplitude method was originally devised by Tikochinsky in 1977, using the quantum theory of scattering. The method is applied to two model potentials, the rectangular potential barrier and the Eckart potential, both with analytical solutions for the reflection coefficient. Numerical results will be compared with the exact values for several energies. The problem of calculating the reflection coefficient, usually involving extensive algebra as described in several textbooks, is reduced to solving a first order differential equation with initial condition. The method is very simple to apply, representing an attractive tool for teaching introductory quantum mechanics. A simple computer code is available from which reflection coefficients for the Eckart potential can be calculated.

We introduce a solution theory for time-varying linear differential-algebraic equations (DAEs) E ( t ) x ˙ = A ( t ) x E(t)\dot x=A(t)x which can be transformed into standard canonical form (SCF); i.e., the DAE is decoupled into an ODE z ˙ 1 = J ( t ) z 1 \dot z_1 = J(t)z_1 and a pure DAE N ( t ) z ˙ 2 = z 2 N(t) \dot z_2 = z_2 , where N N is pointwise strictly lower triangular. This class is a time-varying generalization of time-invariant DAEs where the corresponding matrix pencil is regular. It will be shown in which sense the SCF is a canonical form, that it allows for a transition matrix similar to the one for ODEs, and how this can be exploited to derive a variation of constants formula. Furthermore, we show in which sense the class of systems transferable into SCF is equivalent to DAEs which are analytically solvable, and relate SCF to the derivative array approach, differentiation index and strangeness index. Finally, an algorithm is presented which determines the transformation matrices which put a DAE into SCF.

Bifurcations of dynamical systems described byseveral second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to the occurrence of a hyperbolic chaotic invariant set.

The generalized (n + 1)-dimensional KP equation with variable coefficients is investigated in this paper. The bilinear form of the equation has been obtained by the Hirota direct method. In addition, with the help of Wronskian technique and the Pfaffian properties, Wronskian and Grammian solutions have been generated.

We exhibit an alternative method for solving inhomogeneous second-order linear ordinary dynamic equations on time scales, based on reduction of order rather than variation of parameters. Our form extends recent (and long-standing) analysis on to a new form for difference equations, quantum equations and arbitrary dynamic equations on time scales.

A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.

An efficient self-contained trajectory optimization software is generated by making use of de Pontecoulant's analytic lunar theory removing the need for an outside third body ephemeris program to compute the lunar and solar position vectors at each integration step. The accelerations being further resolved along the rotating Euler–Hill frame after expansion to third order in the spacecraft radial distance, the adjoint differential equations are derived in a direct manner complementing the generation of the dynamic system of equations for full compatibility. Because the variation of parameters equations are cast in terms of the nonsingular equinoctial elements with the perturbation accelerations resolved in their analytic form along the rotating axes, the adjoint equations are also derived in the same manner providing a highly efficient and accurate system of equations for rapid computations in conjunction with Aerospace Corporation's NLP2 nonlinear programming codes to search for the initial values of the multipliers that steer the spacecraft towards its target orbit in minimum time. Numerical simulations show that the solutions obtained by the analysis developed in this paper are essentially identical to the more indirect approach based on the use of inertial accelerations obtained from a separate ephemeris generator and subsequent conversions to the thrust frame and equinoctial system.

We introduce the principal matrix solution Z(t, s) of the linear Volterra-type vector integro-dynamic equation and prove that it is the unique matrix solution of We also show that the solution of is unique and given by the variation of parameters formula

In this paper, a new boundary element analysis for two-dimensional (2D) transient elastodynamic problems is proposed. The dual reciprocity method (DRM) is reconsidered by employing new radial basis functions (RBFs) to approximate the domain inertia terms. These new RBFs, which are in the form of ζ+κ sin (ωr+α), are called Fourier RBFs hereafter. Using the method of variation of parameters, the particular solution kernels of Fourier RBFs corresponding to displacement and traction, whose a few terms are singular, has been explicitly derived. Therefore, a new simple smoothing trick has been employed to resolve the singularity problem. Moreover, the limiting values of the particular solution kernels have been evaluated. In order to find the unknown parameters of Fourier RBFs, an optimization problem seeking for the optimum value of the Houbolt scheme parameter β that minimizes the mean squared error (MSE) function of the problem is established. Since the MSE function of the proposed RBFs is a function of five unknown parameters (i.e., ζ, κ, ω, α, and β), the genetic algorithm (GA) has been used to solve the necessary optimization problem. In order to illustrate the validity, accuracy, and superiority of the present study, several numerical examples are examined and compared to the results of analytical and other RBFs reported in the literature. Compared to other RBFs, Fourier RBFs show more accurate and stable results. Moreover, these results are obtained using less degree of freedom without any additional internal points that are commonly used to improve the accuracy of the results.

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.