scholarly journals The Novel Integral Homotopy Expansive Method

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1204
Author(s):  
Uriel Filobello-Nino ◽  
Hector Vazquez-Leal ◽  
Jesus Huerta-Chua ◽  
Jaime Ramirez-Angulo ◽  
Darwin Mayorga-Cruz ◽  
...  
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
The Novel ◽  
Practical Applications ◽  
Linear And Nonlinear ◽  
Scientific Phenomena ◽  
Analytical Expressions

This work proposes the Integral Homotopy Expansive Method (IHEM) in order to find both analytical approximate and exact solutions for linear and nonlinear differential equations. The proposal consists of providing a versatile method able to provide analytical expressions that adequately describe the scientific phenomena considered. In this analysis, it is observed that the proposed solutions are compact and easy to evaluate, which is ideal for practical applications. The method expresses a differential equation as an integral equation and expresses the integrand of the equation in terms of a homotopy. As a matter of fact, IHEM will take advantage of the homotopy flexibility in order to introduce adjusting parameters and convenient functions with the purpose of acquiring better results. In a sequence, another advantage of IHEM is the chance to distribute one or more of the initial conditions in the different iterations of the proposed method. This scheme is employed in order to introduce some additional adjusting parameters with the purpose of acquiring accurate analytical approximate solutions.

scholarly journals A New Approach to Approximate Solutions for Nonlinear Differential Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Safia Meftah
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Perturbation Method ◽  
Periodic Solutions ◽  
Asymptotic Expansions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Approximation Methods ◽  
New Approach

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

scholarly journals The Laplace-Adomian-Pade Technique for the ENSO Model

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Yi Zeng
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
High Accuracy ◽  
Padé Method

The Laplace-Adomian-Pade method is used to find approximate solutions of differential equations with initial conditions. The oscillation model of the ENSO is an important nonlinear differential equation which is solved analytically in this study. Compared with the exact solution from other decomposition methods, the approximate solution shows the method’s high accuracy with symbolic computation.

Application of Ambarzumian’s Method to Radiant Interchange in a Rectangular Cavity

1974 ◽  
Vol 96 (2) ◽  
pp. 191-196 ◽  
Author(s):  
A. L. Crosbie ◽  
T. R. Sawheny
Keyword(s):  
Differential Equation ◽  
Heat Flux ◽  
Integral Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Integral Equation ◽  
Initial Conditions ◽  
Rectangular Cavity ◽  
Wide Range ◽  
First Time

Ambarzumian’s method had been used for the first time to solve a radiant interchange problem. A rectangular cavity is defined by two semi-infinite parallel gray surfaces which are subject to an exponentially varying heat flux, i.e., q = q0 exp(−mx). Instead of solving the integral equation for the radiosity for each value of m, solutions for all values of m are obtained simultaneously. Using Ambarzumian’s method, the integral equation for the radiosity is first transformed into an integro-differential equation and then into a system of ordinary differential equations. Initial conditions required to solve the differential equations are the H functions which represent the radiosity at the edge of the cavity for various values of m. This H function is shown to satisfy a nonlinear integral equation which is easily solved by iteration. Numerical results for the H function and radiosity distribution within the cavity are presented for a wide range of m values.

scholarly journals APPROXIMATE SOLUTIONS OF DAMPED NON LINEAR SYSTEM WITH VARYING PARAMETER AND DAMPING FORCE

2019 ◽  
pp. 13-18
Author(s):  
REZAUL KARIM ◽  
PINAKEE DEY ◽  
SOMI AKTER ◽  
MOHAMMAD ASIF AREFIN ◽  
SAIKH SHAHJAHAN MIAH
Keyword(s):  
Approximate Solution ◽  
Dynamical Systems ◽  
Linear System ◽  
Harmonic Balance ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Damping Force ◽  
Kbm Method ◽  
Non Linear System

The study of second-order damped nonlinear differential equations is important in the development of the theory of dynamical systems and the behavior of the solutions of the over-damped process depends on the behavior of damping forces. We aim to develop and represent a new approximate solution of a nonlinear differential system with damping force and an approximate solution of the damped nonlinear vibrating system with a varying parameter which is based on Krylov–Bogoliubov and Mitropolskii (KBM) Method and Harmonic Balance (HB) Method. By applying these methods we solve and also analyze the finding result of an example. Moreover, the solutions are obtained for different initial conditions, and figures are plotted accordingly where MATHEMATICA and C++ are used as a programming language.

Symmetric Arc Solutions of ζ¨ = ζn

1968 ◽  
Vol 35 (3) ◽  
pp. 565-570
Author(s):  
C. P. Atkinson ◽  
B. L. Dhoopar
Keyword(s):  
Differential Equation ◽  
Time Domain ◽  
Complex Variable ◽  
Digital Computer ◽  
Nonlinear Differential Equations ◽  
Time Derivative ◽  
Approximate Solutions ◽  
Complex Variables ◽  
Complex Modes ◽  
The Time Domain

This paper, “Symmetric Arc Solutions of ζ¨ = ζn,” presents periodic solutions of this differential equation relating the complex variable ζ(t) = u(t) + iv(t) and its second time derivative ζ¨ The solutions are called symmetric arc solutions since they form such arcs on the ζ = u + iv-plane. The solutions, ζ(t), are “complex modes” of coupled nonlinear differential equations in the complex variables z1 and z2. Symmetric arc solutions are presented for a range of n from n = 3 to n = 101. Approximate solutions are presented and compared with solutions generated by digital computer. Solutions are presented on the ζ-plane and in the time domain as u(t) and v(t).

scholarly journals A Novel Version of HPM Coupled with the PSEM Method for Solving the Blasius Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Uriel Filobello-Nino ◽  
Hector Vazquez-Leal ◽  
Jesus Huerta-Chua ◽  
Victor Manuel Jimenez-Fernandez ◽  
Mario Alberto Sandoval-Hernandez ◽  
...  
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Linear Part ◽  
Integration Process ◽  
Exponential Functions ◽  
Unknown Parameters ◽  
Approximate Analytical Expression ◽  
Practical Applications ◽  
Homotopy Perturbation ◽  
Blasius Problem

This work studies the nonlinear differential equation that models the Blasius problem (BP) which is of great importance in fluid dynamics. The aim is to obtain an approximate analytical expression that adequately describes the phenomenon considered. To find such approximation, we propose a new method denominated powered homotopy perturbation (PHPM). Unlike HPM algorithm, the successive integration process generated by PHPM will consider zero the constants of integration in each approximation, except the last one. In the same way, PHPM will propose an adequate initial trial function provided of some unknown parameters in such a way that it will not evaluate the initial conditions in the iterations of the process; therefore, this set of parameters will be employed with the purpose of adjusting in the best accurate way the proposed approximation at the final part of the process. As a matter of fact, we will note from this analysis that the proposed solution is compact and easy to evaluate and involves a sum of five exponential functions plus a linear part of two terms, which is ideal for practical applications. With the purpose to get a better approximation, we find useful to combine PHPM with the power series extender method (PSEM) which implies to add to the PHPM solution one rational function with parameters to adjust. From this proposal, we find an approximate solution competitive with others from the literature.

Application of New Homotopy Perturbation Method for Solving Partial Differential Equations

2018 ◽  
Vol 15 (2) ◽  
pp. 500-508 ◽  
Author(s):  
Musa R. Gad-Allah ◽  
Tarig M. Elzaki
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Novel Technique ◽  
Homotopy Perturbation ◽  
Linear And Nonlinear ◽  
Linear Differential

In this paper, a novel technique, that is to read, the New Homotopy Perturbation Method (NHPM) is utilized for solving a linear and non-linear differential equations and integral equations. The two most important steps in the application of the new homotopy perturbation method are to invent a suitable homotopy equation and to choose a suitable initial conditions. Comparing between the effects of the method (NHPM), is given exact solution, and the method (HPM), is given approximate solution, in this paper, we make some instances are provided to prove the ability of the method (NHPM). Show that the method (NHPM) is valid and effective, easy and accurate in solving linear and nonlinear differential equations, compared with the Homotopy Perturbation Method (HPM).

scholarly journals Fractional differential equations and Volterra–Stieltjes integral equations of the second kind

2019 ◽  
Vol 38 (4) ◽  
Author(s):  
Avyt Asanov ◽  
Ricardo Almeida ◽  
Agnieszka B. Malinowska
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Equivalence Relation ◽  
Numerical Experiments ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Stieltjes Integral ◽  
Fractional Differential

Abstract In this paper, we construct a method to find approximate solutions to fractional differential equations involving fractional derivatives with respect to another function. The method is based on an equivalence relation between the fractional differential equation and the Volterra–Stieltjes integral equation of the second kind. The generalized midpoint rule is applied to solve numerically the integral equation and an estimation for the error is given. Results of numerical experiments demonstrate that satisfactory and reliable results could be obtained by the proposed method.

Forced Vibrations of an Elastic Plate Interacting With a Layer of Granular Material

1993 ◽  
Author(s):  
Y. Obodan
Keyword(s):  
Differential Equation ◽  
Granular Material ◽  
Elastic Plate ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Critical Time ◽  
Variable Structure ◽  
Plate Boundaries ◽  
Unilateral Constraints ◽  
Layer Behavior

Abstract A variable-structure system, consisting of a thin horizontal elastic plate, loaded by a heavy layer of granular material is considered. The system is excited by prescribed vertical vibrations of the plate boundaries. An essentially nonlinear vibrational rheological model with unilateral constraints is proposed for simulation of the layer behavior. During the coupled motion stages the layer dynamic response on the interface between plate and granular layer is described by the Volterra integro-operator relationship with nonlinear isochronic force-displacement characteristic. The governing set of equations includes the plate dynamics differential equation, an integro-differential equation of the layer dynamics, and also appropriate boundary and initial conditions. Due to utilization of the Galerkin method and Kroosh approximation, the initial set is reduced to the set of nonlinear differential equations. The latter is integrated numerically. Model identification is carried out by the computer processing of the experimental bending strain histories. Influence of the excitation parameters, height of the layer, and damping on the bending strains is analyzed. Adequate simulation of the layer behavior for one-period modes is achieved with regard to the motion periodicity, the values of the strain peaks, the critical time points and the duration of the main time stages. Analysis of numerical phase trajectories and Poincaré sections predicts transition to stochastic vibration modes and the existence of stable two-period limit cycles, when the maximum forcing acceleration is increased above 3g.

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