Exact Solutions for Initial Value Problems of the Davey-Stewartson I Equation under Some Localized Initial Conditions

1998 ◽  
Vol 67 (4) ◽  
pp. 1157-1162 ◽  
Author(s):  
Tetsu Yajima ◽  
Katsuhiro Nishinari
Keyword(s):  
Exact Solutions ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Initial Value

scholarly journals Finding symmetries by incorporating initial conditions as side conditions

2008 ◽  
Vol 19 (6) ◽  
pp. 701-715 ◽  
Author(s):  
JOANNA GOARD
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Initial Condition ◽  
Initial Value ◽  
Side Condition ◽  
Symmetry Generator ◽  
Infinitesimal Symmetry

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

scholarly journals A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra’s Population Growth Model

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Majid Tavassoli Kajani ◽  
Mohammad Maleki ◽  
Adem Kılıçman
Keyword(s):  
Population Growth ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Initial Value ◽  
Step Size ◽  
Integral Term ◽  
Multiple Step

A new shifted Legendre-Gauss collocation method is proposed for the solution of Volterra’s model for population growth of a species in a closed system. Volterra’s model is a nonlinear integrodifferential equation on a semi-infinite domain, where the integral term represents the effects of toxin. In this method, by choosing a step size, the original problem is replaced with a sequence of initial value problems in subintervals. The obtained initial value problems are then step by step reduced to systems of algebraic equations using collocation. The initial conditions for each step are obtained from the approximated solution at its previous step. It is shown that the accuracy can be improved by either increasing the collocation points or decreasing the step size. The method seems easy to implement and computationally attractive. Numerical findings demonstrate the applicability and high accuracy of the proposed method.

scholarly journals Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Galerkin Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Variable Coefficients ◽  
Solution Technique ◽  
Initial Value ◽  
Spectral Approximations ◽  
Base Functions ◽  
And Performance

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

scholarly journals Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

2021 ◽  
Vol 24 (4) ◽  
pp. 32-39
Author(s):  
Hussein M. Sagban ◽  
◽  
Fadhel S. Fadhel ◽  
Keyword(s):  
Approximate Solution ◽  
Rate Of Convergence ◽  
Initial Value Problem ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

The main objective of this paper is to solve fuzzy initial value problems, in which the fuzziness occurs in the initial conditions. The proposed approach, namely the modified variational iteration method, will be used to find the solution of fuzzy initial value problem approximately and to increase the rate of convergence of the variational iteration method. From the obtained results, as it is expected, the approximate results of the proposed method are more accurate than those results obtained without using the modified variational iteration method.

scholarly journals Initial value problems should not be associated to fractional model descriptions whatever the derivative definition used

2021 ◽  
Vol 6 (10) ◽  
pp. 11318-11329
Author(s):  
Jocelyn SABATIER ◽  
◽  
Christophe FARGES
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Model Analysis ◽  
Fractional Differentiation ◽  
Initial Value ◽  
Validity Of Results ◽  
Fractional Model ◽  
Definition Of

<abstract> <p>The paper shows that the Caputo definition of fractional differentiation is problematic if it is used in the definition of a time fractional model and if initial conditions are taken into account. The demonstration is done using simple examples (or counterexamples). The analysis is extended to the Riemann-Liouville and Grünwald-Letnikov definitions. These results thus question the validity of results produced in the field of time fractional model analysis in which initial conditions are involved.</p> </abstract>

On generalized fractional operators and a gronwall type inequality with applications

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5457-5473 ◽  
Author(s):  
Yassine Adjabi ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equations ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Type Inequality ◽  
Weighted Spaces ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Derivatives

In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

scholarly journals Modified variational iteration method for variant Boussinesq equation

2015 ◽  
Vol 19 (4) ◽  
pp. 1195-1199 ◽  
Author(s):  
Jun-Feng Lu
Keyword(s):  
Exact Solutions ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Boussinesq Equation ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

In this paper, we solve the variant Boussinesq equation by the modified variational iteration method. The approximate solutions to the initial value problems of the variant Boussinesq equation are provided, and compared with the exact solutions. Numerical experiments show that the modified variational iteration method is efficient for solving the variant Boussinesq equation.

Evolution of near-inertial waves

1995 ◽  
Vol 301 ◽  
pp. 269-294 ◽  
Author(s):  
R. C. Kloosterziel ◽  
P. Müller
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Fluid Layer ◽  
Normal Modes ◽  
Buoyancy Frequency ◽  
Infinite Domain ◽  
Initial Value ◽  
Flux Boundary Condition

The three-dimensional evolution of near-inertial internal gravity waves is investigated for the case of a laterally unbounded fluid layer of constant finite depth. A general Green's function formulation is derived which can be used to solve initial value problems or study the effect of forcing. The Green's function is expanded in vertical normal modes, and is very singular. Convolutions with finite-sized initial conditions lead however to well-behaved solutions. Expansions in similarity solutions of the diffusion equation are shown to be an alternative for finding exact solutions to initial value problems, with respect to one normal mode. For the case of constant buoyancy frequency normal modes expansions are shown to be equivalent to expansions in an alternative series of which the first term is the response on the infinite domain, all the others being corrections to account for the no-flux boundary condition on the upper and lower boundaries.

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