Solution Of Initial Value Problem Research Articles

First Order Differential Operators Associated to the Space of Monogenic Functions in Parameter-Depending Clifford Algebras

This paper is aimed at finding all linear first order partial differential operators \({\mathcal{F}}\) with parameter-depending Clifford-algebra-valued coefficients, that are associated to the generalized Cauchy-Riemann operator. In order to obtain the conditions on the coefficients of \({\mathcal{F}}\), a Leibniz rule for functions with values in a more general Clifford-type algebra is proved. Using the theory of associated spaces, we show the construction of solutions of initial value problems involving these operators.

On a New Inverse Polynomial Numerical Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations

On a New Inverse Polynomial Numerical Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations

Some Notes on Taylors Series Expansion with ODE´S

In this paper, our aim is to study the numerical solution of initial value problems (IVPs) for ODE´s is one of the fundamental problems in scientific computation. There are many well-established algorithms for approximate solution of IVPs. However, traditional integration methods usually provide only approximate values for the solution. Precise error bounds are rarely available the error estimates, which are sometimes delivered, are not guaranteed to be accurate and are sometimes unreliable. The main goal of the paper is to present all the existent approaches together emphasizing that since, all researches face the same problem but in different contexts, they are finding the same kinds of problems in spite of there different for mails m5 and methodologies.

Complexity of parametric initial value problems for systems of ODEs

We study the approximate solution of initial value problems for parameter dependent finite or infinite systems of scalar ordinary differential equations (ODEs). Both the deterministic and the randomized setting is considered, with input data from various smoothness classes. We study deterministic and Monte Carlo multilevel algorithms and derive convergence rates. Moreover, we prove their optimality by showing matching (in some limit cases up to logarithmic factors) lower bounds and settle this way the complexity. Comparisons between the deterministic and randomized setting are given, as well.

Projection-Grid Method of Elasticity Problems Solution in Flight Dynamics

Background. The development of efficient projection-grid method for solving initial-boundary value problems of the elastic dynamics of the aircraft. Objective. Theoretical study of numerical methods for solving the elastic dynamics of aircraft in order to create a generalized method of numerical integration of initial-boundary value problems for discrete-continuum. Methods. As a generalized mathematical description of the initial-boundary value problem by using the operator formulation of the first order main part. Approximate solution of initial value problems of elastic dynamics of aircraft represented as a linear form on the class of admissible functions of non-degenerate projective basis. Algebraization of the spatial variables is realized because of orthogonalization residuals of equations and boundary conditions for the system of functions defining non-degenerate weight basis. The greatest effect is achieved by computing the matching item in the projection and a weight basis in conjunction with the weak formulation of the Galerkin method in the form of the finite element method. The general form of the finite difference method is used for algebraization of unknown functions on a temporary argument. For solving systems of nonlinear algebraic equations on time layers, Newton's method and its modifications were applied. Results. A general approach to solving the problems of the elastic dynamics of aircraft using the procedure of algebraization based on projection-grid schemes of the method of weighted residuals. A posteriori estimates for the accuracy, convergence and stability of numerical solutions of the elastic dynamics of aircraft were presented. Conclusions. The developed technique of algebraization tasks in elastic dynamics of aircraft can be widely used in the simulation of the dynamics of liquid carrier rockets in different parts of the flight.

General Nyström methods in Nordsieck form: Error analysis

The paper is concerned with the analysis of the error associated to a family of multi-value numerical methods for the solution of initial value problems based on special second order ordinary differential equations. Such methods, denoted as General Nyström methods, provide at each step point an approximation to the Nordsieck vector associated to the solution of the problem. Order issues for such methods based on the theory of rooted trees are here provided, as well as an accuracy analysis is carried out, leading to a representation of the local truncation error.

The initial value problem for interval-valued second-order differential equations under generalized H-differentiability

In this paper the interval-valued second-order differential equations under generalized Hukuhara differentiability (ISDEs) are introduced. Under suitable conditions we obtain the existence and uniqueness results of solutions to ISDEs. To prove this assertion we use idea of contraction principle and successive approximations. Furthermore, we use the method based on properties of linear transformations (LTM) to find the explicit solution of initial value problem for linear second-order differential equation with interval-valued forcing function and with interval initial values (IIVP). We apply the linear transformations method to two example problems including a vibrating mass and an electrical circuit. Finally, the method based on the analysis of a solution to real-valued second-order differential equation is investigated to solve interval-valued second-order differential equations with interval-valued coefficients, interval-valued forcing function and interval initial values. Some examples are presented to illustrate applicability of the proposed method.

Modified spline collocation for linear fractional differential equations

We propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. Theoretical results are verified by some numerical examples.

Integrable solutions for implicit fractional order functional differential equations with infinite delay

In this paper we study the existence of integrable solutions for initial value problem for implicit fractional order functional differential equations with infinite delay. Our results are based on Schauder type fixed point theorem and the Banach contraction principle fixed point theorem.

A class of the backward Euler's method for initial value problems

In this paper, we propose a class of the backward Euler's method for the numerical solution of initial value problems of ordinary differential equations. The proposed class is constructed by considering a suitable interpolating function. The accuracy and stability of the proposed class are considered. A comparison of numerical results obtained by some of the proposed methods and the existing classical backward Euler's method is given which demonstrate that for the problems tested here the proposed methods outperform the existing classical method.

A Comparative Study on Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Runge Kutta Methods

This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac-curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed methods for different step sizes to examine superiority. Several numerical examples are given to demonstrate the reliability and efficiency.

A digraph theoretic parallelism in block methods

In the exploitation of sequential computation expertise in the parallel solution of initial value problems in ordinary differential equations three notable directions have evolved, these are listed to include, parallelism across the method, parallelism across the steps and parallelism across the system. In this paper, parallel block methods which are methods of the first approach above are considered to take advantage of parallelism across the machine by employing the theoretic concepts of digraphs. This has been used by Jackson, Norsett and Iserles in their search for parallelism in Runge–Kutta Methods (RKM). This approach presents a higher prospective advantage for block methods arising from the fact that for a given block dimension, a block method can attain higher order and may have a wider stability region over a RKM with a comparable number of stages. Furthermore, the notion of isomorphic methods are introduced, an example of which is the embedded RKM pairs, more are presented in our generalized formulation of multi-block methods.

Comparison of three unsupervised neural network models for first Painlevé Transcendent

In this paper, a reliable soft computing framework is presented for the approximate solution of initial value problem (IVP) of first Painleve equation using three unsupervised neural network models optimized with sequential quadratic programming (SQP). These mathematical models are constructed in the form of feed-forward architecture including log-sigmoid, radial base and tan-sigmoid activation functions in the hidden layers. The optimization of designed parameters for each model is performed with SQP, an efficient constraint optimization problem-solving algorithm. The designed methodology is tested on the IVP, and comparative study is carried out with standard solution based on numerical and analytical solvers. The accuracy, convergence and effectiveness of the schemes are validated on the given benchmark problem by large number of simulations and their comprehensive analysis.

Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation

In this paper, novel computing approach using three different models of feed-forward artificial neural networks (ANNs) are presented for the solution of initial value problem (IVP) based on first Painlevé equation. These mathematical models of ANNs are developed in an unsupervised manner with capability to satisfy the initial conditions exactly using log-sigmoid, radial basis and tan-sigmoid transfer functions in hidden layers to approximate the solution of the problem. The training of design parameters in each model is performed with sequential quadratic programming technique. The accuracy, convergence and effectiveness of the proposed schemes are evaluated on the basis of the results of statistical analyses through sufficient large number of independent runs with different number of neurons in each model as well. The comparisons of these results of proposed schemes with standard numerical and analytical solutions validate the correctness of the design models.

Stationary response of nonlinear system with Caputo-type fractional derivative damping under Gaussian white noise excitation

This paper aimed to investigate the response of single-degree-of-freedom system with Caputo-type fractional derivative damping term under Gaussian white noise excitation. First, the approximately analytical solution of the system is obtained using the stochastic averaging method. Then, an effective algorithm for the solution of initial value problems with Caputo derivative is briefly introduced. At last, in order to certify the validity of the analytical solution, two examples are worked out in detail. A very satisfactory agreement is found between the analytical results and the Monte Carlo simulation of original systems.

Fixed point theorems for φ-contractions

This paper deals with the fixed point theorems for mappings satisfying a contractive condition involving a gauge function φ when the underlying set is endowed with a b-metric. Our results generalize/extend the main results of Proinov and thus we obtain as special cases some results of Mysovskih, Rheinboldt, Gel’man, and Huang. We also furnish an example to substantiate the validity of our results. Subsequently, an existence theorem for the solution of initial value problem has also been established.

Approximate Solution of Initial Value Problems by Means of Ninth Degree Spline

In this paper, a nonlinear initial value problem is solved numerically by means of ninth
 degree spline function. The solution of initial value problems approximated as a linear
 interpolation of ninth spline functions. In this method, the basis spline functions are redefined into
 a new approximation set of ninth degree spline functions which interpolate the number of select
 derivatives. To test the efficiency of the method, two numerical examples of initial value problems
 are solved by the proposed method.

Approximating solutions of nonlinear hybrid differential equations

Abstract We prove the existence as well as approximations of the solutions of initial value problems of first order ordinary nonlinear hybrid differential equations. We rely our results on a recent hybrid fixed point theorem of Dhage (2014) in partially ordered normed linear spaces. A realization is also indicated to illustrate the abstract theory developed in the paper.

Runge–Kutta projection methods with low dispersion and dissipation errors

In this paper new one-step methods that combine Runge---Kutta (RK) formulae with a suitable projection after the step are proposed for the numerical solution of Initial Value Problems. The aim of this projection is to preserve some first integral in the numerical integration. In contrast with standard orthogonal projection, the direction of the projection at each step is obtained from another suitable embedded formula so that the overall method is affine invariant. A study of the local errors of these projection methods is carried out, showing that by choosing proper embedded formulae the order can be increased for the harmonic oscillator. Particular embedded formulae for the third order method by Bogacki and Shampine (BS3) are provided. Some criteria to get appropriate dynamical directions for general problems as well as sufficient conditions that ensure the existence of RK methods embedded in BS3 according to them are given. Finally, some numerical experiments to test the behaviour of the new projection methods are presented.

On Effective Convergence of Numerical Solutions for Differential Equations

This article studies the effective convergence of numerical solutions of initial value problems (IVPs) for ordinary differential equations (ODEs). A convergent sequence { Y m } of numerical solutions is said to be effectively convergent to the exact solution if there is an algorithm that computes an N ∈ ℕ, given an arbitrary n ∈ ℕ as input, such that the error between Y m and the exact solution is less than 2 -n for all m ≥ N . It is proved that there are convergent numerical solutions generated from Euler’s method which are not effectively convergent. It is also shown that a theoretically-proved-computable solution using Picard’s iteration method might not be computable by classical numerical methods, which suggests that sometimes there is a gap between theoretical computability and practical numerical computations concerning solutions of ODEs. Moreover, it is noted that the main theorem (Theorem 4.1) provides an example of an IVP with a nonuniform Lipschitz function for which the numerical solutions generated by Euler’s method are still convergent.