Solution Of Initial Value Problem Research Articles

Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena. In this paper, we develop an exponentially accurate Jacobi–Gauss–Lobatto collocation (J–GL-C) method to solve the variable-order fractional Schrödinger equations in one dimension (1D) and two dimensions (2D). In this method, the aforementioned problem is reduced to a system of ordinary differential equations (ODEs) in the time variable. As a result, we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations, one based on the implicit Runge–Kutta (IRK) method of fourth order and the other based on Jacobi–Gauss–Radau collocation (J–GR-C) method. The validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1D and 2D. The convergence of the methods is graphically analyzed. The results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable-order nonlinear partial differential equations.

APPROXIMATION OF THE DISTANCE IN THE SPHERE THROUGH OF THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEM ASSOCIATED TO GEODESICS

In this paper, we proposes an algorithm to approximate the Geodesic distance between points p and q of sphere, through the numerical solution of initial value problem associated with the system of ordinary differential equations of the geodesics; for this an appropriate direction is obtained.

On the General Solution of Initial Value Problems of Ordinary Differential Equations Using the Method of Iterated Integrals

Our goal is to give a very simple, effective and intuitive algorithm for the solution of initial value problem of ODEs of 1st and arbitrary higher order with general i.e. constant, variable or nonlinear coefficients. We find a local expansion of the differential equation/function that employs the higher derivatives (in the sense of total derivatives) of the differential equation/function with respect to independent variable to find an approximation which can be successively improved to desired accuracy by adding extra terms. The method can also be used easily for general 2nd and higher order ODEs. We explain with examples the steps to solution of initial value problems of 1st order ODEs and later follow with more examples for linear and non-linear 2nd order ODEs.

A new stepsize change technique for Adams methods

Abstract In this paper a new technique for stepsize changing in the numerical solution of Initial Value Problems for ODEs by means of Adams type methods is considered. The computational cost of the new technique is equivalent to those of the well known interpolation technique (IT). It is seen that the new technique has better stability properties than the IT and moreover, its leading error term is smaller. These facts imply that the new technique can outperform the IT.

Sufficient Conditions for First-Order Differential Operators to be Associated with a q-Metamonogenic Operator in a Clifford Type Algebra

Consider the initial value problem 0.1 $$\begin{aligned} \partial _{t}u= & {} {\mathcal L}(t,x,u,\partial _{x_{i}}u),\nonumber \\ u(0,x)= & {} \varphi (x), \end{aligned}$$ where t is the time, $${\mathcal L}$$ is a linear first-order differential operator and $$\varphi $$ is a generalized q-metamonogenic function. This problem can be solved by applying the method of associated spaces which is constructed by Tutschke (see Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer, New York, 1989). In this work, we formulate sufficient conditions on the coefficients of the operator $${\mathcal L}$$ under which this operator is associated to the space of generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side, when q and $$\lambda $$ are constant Clifford vectors. We also build a computational algorithm to check the computations in the cases $${\mathcal A}^{*}_{2,2}$$ and $${\mathcal A}^{*}_{3,2}$$ . In conical domains, the initial value problem (0.1) is uniquely solvable for an operator $${\mathcal L}$$ and for any generalized q-metamonogenic initial function $$\varphi $$ , provided an interior estimate holds for generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side. The solution is also a generalized q-metamonogenic function for each fixed t. This work generalizes the results given in Di Teodoro and Sapian (Adv. Appl. Clifford Algebras, 25:283–301, 2015) and Van (Differential operator in a Clifford analysis associated to differential equations with anti-monogenic right hand side, IC/2006/134, 2016).

Excitation of unsteady Görtler vortices by localized surface nonuniformities

A combined theoretical and numerical analysis of an experiment devoted to the excitation of Gortler vortices by localized stationary or vibrating surface nonuniformities in a boundary layer over a concave surface is performed. A numerical model of generation of small-amplitude disturbances and their downstream propagation based on parabolic equations is developed. In the framework of this model, the optimal and the modal parts of excited disturbance are defined as solutions of initial-value problems with initial values being, respectively, the optimal disturbance and the leading local mode at the location of the source. It is shown that a representation of excited disturbance as a sum of the optimal part and a remainder makes it possible to describe its generation and downstream propagation, as well as to predict satisfactorily the corresponding receptivity coefficient. In contrast, the representation based on the modal part provides only coarse information about excitation and propagation of disturbance in the range of parameters under investigation. However, it is found that the receptivity coefficients estimated using the modal parts can be reinterpreted to preserve their practical significance. A corresponding procedure was developed. The theoretical and experimental receptivity coefficients are estimated and compared. It is found that the receptivity magnitudes grow significantly with the disturbance frequency. Variation of the span-wise scale of the nonuniformities affects weakly the receptivity characteristics at zero frequency. However, at high frequencies, the efficiency of excitation of Gortler vortices depends substantially on the span-wise scale.

Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices

Probabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: \(n \times n^{j+1}\) matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a Henon–Heiles system is provided.

Piecewise continuous solutions of initial value problems of singular fractional differential equations with impulse effects

Results on the existence of piecewise continuous solutions for two classes of initial value problems of impulsive singular fractional differential equations are obtained.

Existence and uniqueness of global solutions of caputo-type fractional differential equations

Abstract In this paper we consider initial value problems for fractional differential equations involving Caputo differential operators. By establishing a new property of the Mittag-Leffler function and using the Schauder fixed point theorem, we obtain new sufficient conditions for the existence and uniqueness of global solutions of initial value problems.

A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic

A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculation of higher order derivatives formally is an over-elaborate task. To avoid the analytical computation of the successive derivatives, numeric and automatic differentiation are usually used. A recent alternative to these techniques is based on the calculation of higher derivatives by using the Infinity Computer—a new computational device allowing one to work numerically with infinities and infinitesimals. Two variants of a one-step multi-point method closely related to the classical Taylor formula of order three are considered. It is shown that the new formula is order three accurate, though requiring only the first two derivatives of y(t) (rather than three if compared with the corresponding Taylor formula of order three). To get numerical evidence of the theoretical results, a few test problems are solved by means of the new methods and the obtained results are compared with the performance of Taylor methods of order up to four.

Existence and approximate solutions for hybrid fractional integro-differential equations

In this paper we prove existence and approximations of the solutions for initial value problems of non-linear hybrid fractional differential equations, using the operator theoretic technique in a partially ordered metric space proved by Dhage.

The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces

By using the partial order method and some new comparison results, the maximal or minimal solution of the initial value problem for nonlinear second order integro-differential equations with delayed arguments in Banach spaces are investigated. In this paper, we require only a lower solution or an upper solution and some weaker conditions presented here, and we extend and improve some recent results (see [1-11]).

Evolutionary construction of Runge–Kutta–Nyström pairs of orders 5(4)

Runge – Kutta – Nystrom pairs of orders 5 and 4 are well suited for the solution of Initial value problem of second order. During the last decades were pushed aside by the pairs of orders 6 and 4 share the same number cost. Namely, 5 stages per step. Here we propose an alternative case of such pairs using only four stages per step. This was achieved by solving the corresponding equations of condition by the technique of Differential Evolution. Finally some numerical tests justify our effort.

Design and Analysis of Some Third Order Explicit Almost Runge-Kutta Methods

In this paper, we propose two new explicit Almost Runge-Kutta (ARK) methods, ARK3 (a three stage third order method, i.e., s = p = 3) and ARK34 (a four-stage third-order method, i.e., s = 4, p = 3), for the numerical solution of initial value problems (IVPs). The methods are derived through the application of order and stability conditions normally associated with Runge-Kutta methods; the derived methods are further tested for consistency and stability, a necessary requirement for convergence of any numerical scheme; they are shown to satisfy the criteria for both consistency and stability; hence their convergence is guaranteed. Numerical experiments carried out further justified the efficiency of the methods.

A first approach in solving initial-value problems in ODEs by elliptic fitting methods

Exponentially-fitted and trigonometrically-fitted methods have a long successful history in the solution of initial-value problems, but other functions might be considered in adapted methods. Specifically, this paper aims at the derivation of a new numerical scheme for approximating initial value problems of ordinary differential equations using elliptic functions. The example considered is the undamped Duffing equation where the forcing term is of autonomous type affected by a perturbation parameter. The new scheme is constructed by considering a suitable approximation to the theoretical solution based on elliptic functions. The proposed elliptic fitting procedure has been tested on a variety of problems, showing its good performance.

Local Bifurcations Analysis of a State-Dependent Delay Differential Equation

In this paper, a first-order equation with state-dependent delay and with a nonlinear right-hand side is considered. Conditions of existence and uniqueness of the solution of initial value problem are supposed to be executed. The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. Local dynamics depends on real parameters which are coefficients of equation right-hand side decomposition in a Taylor series. The parameter which is a coefficient at the linear part of this decomposition has two critical values which determine a stability domain of zero equilibrium. We introduce a small positive parameter and use the asymtotic method of normal forms in order to investigate local dynamics modifications of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the first of these critical values, and the supercritical Andronov – Hopf bifurcation occurs near the second of them (if the sufficient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered as an example. The bifurcation parameter of this equation has one critical value. A simple sufficient condition of Andronov – Hopf bifurcation occurence in the considered equation near a critical value is obtained as a result of applying the method of normal forms.

Construction of measures of noncompactness of DC n [ J , E ] $\mathit{DC}^{n}[J,E]$ and C 0 n [ J , E ] $C^{n}_{0}[J,E]$ with application to the solvability of nth-order integro-differential equations in Banach spaces

In the present paper, we first investigate the construction of compact sets of $\mathit{DC}^{n}[J,E]$ and $C^{n}_{0}[J,E]$ , and then we introduce new measures of noncompactness on these spaces. In addition, as an application, we discuss the existence of solutions of initial value problems for nth-order nonlinear integro-differential equations of mixed type on an infinite interval in Banach spaces. We will also state an interesting example which shows that our results can apply for solving infinite systems of integro-differential equations.

APPROXIMATION OF THE DISTANCE IN THE PLANE THROUGH THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEMS ASSOCIATED TO GEODESICS

In this paper, we proposes an algorithm to approximate the Euclidean distance between points p and q of plane, by doing a search of geodesic that depart from the point p and arrive to a neighborhood of the point q . This search is done through the numerical solution of initial value problems associated with the system of ordinary differential equations of the geodesics; for this is choose a set of directions in the plane.

Extending high order derivatives for special differential equations of the form \(y' = f(y)\) by using monotonically labeled rooted trees.

This paper presents a review of the role played by labeled rooted trees to obtain derivatives for numerical solution of initial value problems in special case \(y' = f(y), y(x_0) = y_0\). We extend a process to find successive derivatives according to monotonically labeled rooted trees, and prove some relevant lemmas and theorems. In this regard, the  derivatives, of the monotonically labeled rooted trees with n vertices are presented by using the monotonically labeled rooted trees with k + n vertices. Eventually, this process is applied to trees without labeling.

Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the $C^{\mathbf {k}}$ category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems and give a number of computer assisted proofs in the analytic framework.