Solution Of Initial Value Problem Research Articles

On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation

In all numerical methods, it is necessary to ascertain the validity of any particular scheme. And this is possible to determine, by verifying the nature of the stability of that scheme. So the general stability function definition is given, from where an investigation is carried out on a class of rational integrator of order 15, to establish the region of absolute stability of the scheme, by constructing the Jordan curve. In the process of expanding the rational function, binomial theorem as well as the idea of combination process were introduced to ease the computation by using Maple-18 package. The simplification of the general rational integrator formula, is constructed from two processes namely through complex function, and then through polar analysis, The Jordan curve is constructed with the help of MATLAB package. Furthermore, it was discovered that the region of instability is on the positive side of the complex plane, while the region of absolute stability is outside the Jordan curve. Finally, it is further established that the encroachment point, τ, lie within the interval ± 140.6. And the encroachment point is visible from the corresponding values of ø and R at the extremes. The stability curve revealed that the integrator is not only A-stable, but also L- stable.

Non-uniqueness of solution for initial value problem of impulsive Caputo-Katugampola fractional differential equations

Non-uniqueness of solution for initial value problem of impulsive Caputo-Katugampola fractional differential equations

Asymptotics behavior for the integrable nonlinear Schrödinger equation with quartic terms: Cauchy problem

We consider the Cauchy problem of integrable nonlinear Schrödinger equation with quartic terms on the line. The first part of the paper considers the Riemann-Hilbert formula via the unified method(also known as the Fokas method). The second part of the paper establishes asymptotic formulas for the solution of initial value problem using the nonlinear steepest descent method(also known as the Deift-Zhou method).

Explicit Solutions of Initial Value Problems for Linear Scalar Riemann-Liouville Fractional Differential Equations With a Constant Delay

In this paper, we study Linear Riemann-Liouville fractional differential equations with a constant delay. The initial condition is set up similarly to the case of ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.

Positive solution of nonlinear fractional differential equations with Caputo‐like counterpart hyper‐Bessel operators

By studying a weakly singular integral whose kernel involves Mittag‐Leffler functions, we obtain some new Gronwall‐type integral inequalities. Applying these inequalities and fixed point theorems, existence and uniqueness of positive solution of initial value problem to nonlinear fractional differential equation with Caputo‐like counterpart hyper‐Bessel operators are established.

The Group-Theoretical Analysis of Nonlinear Optimal Control Problems with Hamiltonian Formalism

In this study, we pay attention to novel explicit closed-form solutions of optimal control problems in economic growth models described by Hamiltonian formalism by utilizing mathematical approaches based on the theory of Lie groups. For this analysis, the Hamiltonian functions, which are used to define an optimal control problem, are considered in two different types, namely, the current and present value Hamiltonians. Furthermore, the first-order conditions (FOCs) that deal with Pontrygain maximum principle satisfying both Hamiltonian functions are considered. FOCs for optimal control in the problem are studied here to deal with the first-order coupled systems. This study mainly focuses on the analysis of these systems concerning for to the theory of symmetry groups and related analytical approaches. First, Lie point symmetries of the first-order coupled systems are derived, and then by using the relationships between symmetries and Jacobi last multiplier method, the first integrals and corresponding invariant solutions for two different economic models are investigated. Additionally, the solutions of initial-value problems based on the transversality conditions in the optimal control theory of economic growth models are analyzed.

C-class functions in generalized metric spaces and applications

Some coincidence and common fixed points are established in G-metric spaces via C-class functions covering an extensive class of contractive conditions. Our outcomes do not rely on the completeness of space/subspace, the continuity or the containment of range space of included tangential pair of single valued mappings. Some interesting examples are furnished and the solutions of initial value problems are given to demonstrate the usability of results acquired.

Global existence theory for fractional differential equations: New advances via continuation methods for contractive maps

Abstract The aim of this article is to form new existence theory for global solutions to nonlinear fractional differential equations. Traditional approaches to existence, uniqueness and approximation of global solutions for initial value problems involving fractional differential equations have been unwieldy or intractable due to the limitations of previously used methods. This includes, for example, certain invariance conditions of the underlying local fixed point strategies. Herein we draw on an alternative tactics, applying the more modern ideas of continuation methods for contractive maps to fractional differential equations. In doing so, we shed new light on the situation, producing these new perspectives through a range of novel theorems that involve sufficient conditions under which global existence, uniqueness, approximation and location of solutions are ensured.

Numerical solution of initial value problems by rational interpolation method using Chebyshev polynomials

In this research, a modified rational interpolation method for the numerical solution of initial value problem is presented. The proposed method is obtained by fitting the classical rational interpolation formula in Chebyshev polynomials leading to a new stability function and new scheme. Three numerical test problems are presented in other to test the efficiency of the proposed method. The numerical result for each test problem is compared with the exact solution. The approximate solutions are show competitiveness with the exact solutions of the ODEs throughout the solution interval.Keywords and Phrases: Chebyshev polynomial, Rational Interpolation, Minimaxpolynomial, Initial Value Problems and Ordinary Differential Equations (ODEs)

Further Nonlinear Retarded Integral Inequalities for Gronwall–Bellman Type and Their Applications

In this article, we will derive several new nonlinear retarded integral inequalities for Gronwall–Bellman type and then apply them as handy tools to examine explicit bounds for solutions of initial value problems, and will also use to evaluate the qualitative as well as the quantitative behavior of solutions of nonlinear retarded differential and integral equations. These inequalities extend some present inequalities to the current literature. An application is also given to demonstrate the strength of our results.

Theory of Hybrid Fractional Differential Equations with Complex Order

We develop the theory of hybrid fractional differential equations with the complex order $thetain mathbb{C}$, $theta=m+ialpha$, $0<mleq 1$, $alphain mathbb{R}$, in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is $mathfrak{D}$- Lipschitzian and the other one is completely continuous, we prove the existence of mild solutions of initial value problems for hybrid fractional differential equations. Finally, an application to solve one-variable linear fractional Schrodinger equation with complex order is given.

Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients

This paper proposes a series-representations for the solution of initial value problems of linear inhomogeneous fractional differential equation with continuous variable coefficients. It is proved that the solution of the problem is determined by adding the solution of the inhomogeneous differential equations with the homogeneous initial conditions to the linear combination of the canonical fundamental system of solution for corresponding homogeneous fractional differential equation and the inhomogeneous initial values. The effectiveness of the theoretical analysis is illustrated with two examples.

Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation

In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order 0&lt;α&lt;1 is considered. The uniqueness of solution is established by using the contraction mapping principle and the existence of solution is obtained by Schauder fixed point theorem.

An Adaptive Step Implicit Midpoint Rule for the Time Integration of Newton\u2019s Linearisations of Non-Linear Problems with Applications in Micromagnetics

The implicit mid-point rule is a Runge–Kutta numerical integrator for the solution of initial value problems, which possesses important properties that are relevant in micromagnetic simulations based on the Landau–Lifshitz–Gilbert equation, because it conserves the magnetization length and accurately reproduces the energy balance (i.e. preserves the geometric properties of the solution). We present an adaptive step size version of the integrator by introducing a suitable local truncation error estimator in the context of a predictor-corrector scheme. We demonstrate on a number of relevant examples that the selected step sizes in the adaptive algorithm are comparable to the widely used adaptive second-order integrators, such as the backward differentiation formula (BDF2) and the trapezoidal rule. The proposed algorithm is suitable for a wider class of non-linear problems, which are linearised by Newton’s method and require the preservation of geometric properties in the numerical solution.

Projector Approach to Constructing Asymptotic Solution of Initial Value Problems for Singularly Perturbed Systems in Critical Case

Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix A ( t ) standing in front of the unknown function. In the present paper, the orthogonal projectors onto k e r A ( t ) and k e r A ( t ) ′ (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction.

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

AbstractEntropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.

On the Use of Aboodh Transform for Solving Non-integer Order Dynamical Systems

In this research study, a transform technique called the Aboodh transform has been employed to find closed form solutions of initial value problems in non-integer order ordinary differential equations. A newly derived theorem in the present study provides a general structure for finding the exact solutions of such non-integer order problems to be solved by this transform technique. Upon comparison with existing transforms including Laplace, Mellin, Sumudu and Elzaki, the Aboodh transform can be found in good agreement. Various non-integer order differential equations including Bagley-Torvik type equation have successfully been solved by the Aboodh transform technique.

Representation of solution of initial value problem for fuzzy linear multi-term fractional differential equation with continuous variable coefficient

We consider the representation of solutions of the initial value problems of fuzzy linear multi-term in-homogeneous fractional differential equations with continuous variable coefficients.

Compacton Solutions of the Korteweg–de Vries Equation with Constrained Nonlinear Dispersion

The numerical solution of initial value problems is used to obtain compacton and kovaton solutions of K(f m, g n) equations generalizing the Korteweg–de Vries K(u2, u1) and Rosenau–Hyman K(u m, u n) equations to more general dependences of the nonlinear and dispersion terms on the solution u. The functions f(u) and g(u) determining their form can be linear or can have the form of a smoothed step. It is shown that peakocompacton and peakosoliton solutions exist depending on the form of the nonlinearity and dispersion. They represent transient forms combining the properties of solitons, compactons, and peakons. It is shown that these solutions can exist against an inhomogeneous and nonstationary background.

Hybrid Numerical Method with Block Extension for Direct Solution of Third Order Ordinary Differential Equations

This paper focuses on the development of a hybrid method with block extension for direct solution of initial value problems (IVPs) of general third-order ordinary differential equations. Power series was used as the basis function for the solution of the IVP. An approximate solution from the basis function was interpolated at some selected off-grid points while the third derivative of the approximate solution was collocated at all grid and off-grid points to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was implemented with five test problems including the Genesio equation to confirm its accuracy and usability. The rate of convergence (ROC) reveals that the method is consistent with the theoretical order of the proposed method. Comparison of the results with some existing methods shows the superiority of the accuracy of the method.