scholarly journals A Picard-Maclaurin theorem for initial value PDEs

2000 ◽  
Vol 5 (1) ◽  
pp. 47-63 ◽  
Author(s):  
G. Edgar Parker ◽  
James S. Sochacki
Keyword(s):  
Differential Equation ◽  
Series Solution ◽  
Initial Conditions ◽  
Broad Class ◽  
Picard Iteration ◽  
Power Series Solution ◽  
Initial Value ◽  
Arbitrary Degree ◽  
Cauchy Theorem ◽  
Analogous Theorem

In 1988, Parker and Sochacki announced a theorem which proved that the Picard iteration, properly modified, generates the Taylor series solution to any ordinary differential equation (ODE) onℜnwith a polynomial generator. In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. Since the domain of a solution of a PDE is a subset ofℜn, we identify one component of the domain to achieve the analogy with ODEs. The generator for the PDE must be a polynomial and autonomous with respect to this component, and no partial derivative with respect to this component can appear in the domain of the generator. The initial conditions must be given in the designated component at zero and must be analytic in the nondesignated components. The power series solution of such a PDE, whose existence is guaranteed by the Cauchy theorem, can be generated to arbitrary degree by Picard iteration. As in the ODE case these conditions can be met, for a broad class of PDEs, through polynomial projections.

scholarly journals Matrix-valued Bratu equation and the exact solution of its initial value problem

2020 ◽  
pp. 1950007
Author(s):  
Hiroto Inoue
Keyword(s):  
Exact Solution ◽  
Power Series ◽  
Initial Value Problem ◽  
Series Solution ◽  
Power Series Solution ◽  
Matrix Solution ◽  
Initial Value ◽  
Bratu Equation ◽  
Exponential Matrix

A matrix-valued extension of the Bratu equation is defined. For its initial value problem, the exponential matrix solution and power series solution are provided.

scholarly journals Similarity Solutions for Multiterm Time-Fractional Diffusion Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
A. Elsaid ◽  
M. S. Abdel Latif ◽  
M. Maneea
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Series Solution ◽  
Error Function ◽  
Fractional Diffusion ◽  
Similarity Solutions ◽  
Fractional Diffusion Equation ◽  
Power Series Solution ◽  
Time Fractional Diffusion Equation ◽  
Similarity Method

Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval(0,1]and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on the obtained results, we propose a definition for a multiterm error function with generalized coefficients.

Analytical and numerical solutions of generalized Burgers' equation via Buckingham's Pi-theorem

2005 ◽  
Vol 83 (10) ◽  
pp. 1035-1049
Author(s):  
I A Hassanien ◽  
A A Salama ◽  
H A Hosham
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Similarity Solutions ◽  
Initial Value ◽  
Generalized Burgers Equation ◽  
One Step ◽  
Pi Theorem ◽  
Scaling Invariant

A generalized dimensional analysis performed by using Buckingham's Pi-theorem for the generalized Burgers' equation is presented. The application of the Buckingham Pi-theorem is used to reduce the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with appropriate corresponding conditions. By using a scaling invariant we simplify the similarity solutions, which are discussed for a specific choice of boundary conditions, and yield analytical solutions, which are in closed form. Also, using extended one-step methods of order five we solve the final ordinary differential equations. This criterion for solvability involves converting the boundary value problem to an initial value problem. PACS Nos.: 02.60.Lj, 47.27.Jv

Global solutions of linear ordinary differential equations with polynomial coefficients

1975 ◽  
Vol 344 (1636) ◽  
pp. 51-64 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Power Series ◽  
Series Solution ◽  
Global Solutions ◽  
Linear Ordinary Differential Equation ◽  
Power Series Solution ◽  
Constructive Approach ◽  
Linear Ordinary Differential Equations ◽  
Polynomial Coefficients

A constructive approach is given, closely based on the work of Ford (1936) for continuing analytically a power series solution of a linear ordinary differential equation with polynomial coefficients outside the circle of convergence.

scholarly journals On Perturbative Cubic Nonlinear Schrodinger Equations under Complex Nonhomogeneities and Complex Initial Conditions

2009 ◽  
Vol 2009 ◽  
pp. 1-29
Author(s):  
Magdy A. El-Tawil ◽  
Maha A. El-Hazmy
Keyword(s):  
Approximate Solution ◽  
Eigenfunction Expansion ◽  
Series Solution ◽  
Initial Conditions ◽  
Solution Algorithm ◽  
Power Series Solution ◽  
Nonlinear Case ◽  
Nonlinear Schrödinger ◽  
Method Of Solution ◽  
General Complex

A perturbing nonlinear Schrodinger equation is studied under general complex nonhomogeneities and complex initial conditions for zero boundary conditions. The perturbation method together with the eigenfunction expansion and variational parameters methods are used to introduce an approximate solution for the perturbative nonlinear case for which a power series solution is proved to exist. Using Mathematica, the symbolic solution algorithm is tested through computing the possible approximations under truncation procedures. The method of solution is illustrated through case studies and figures.

scholarly journals POLYNOMIAL SERIES SOLUTION OF BENJAMIN−BONA−MAHONY EQUATION VIA DIFFERENTIAL TRANSFORM METHOD

2020 ◽  
Vol 4 (1) ◽  
pp. 01-03
Author(s):  
Muhammad Jamil ◽  
Badshah- E-Room
Keyword(s):  
General Solution ◽  
Initial Value Problem ◽  
Series Solution ◽  
Initial Conditions ◽  
Differential Transform Method ◽  
Initial Value ◽  
Transform Method ◽  
Differential Transform ◽  
Polynomial Series

In this work, we solved the initial value problem of Benjamin−Bona−Mahony equation with generalized initial conditions by using differential transform method (DTM). We obtained the general solution in the form of a series. An example is presented for the particular initial conditions.

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