scholarly journals Approximate Solutions for Fractional Boundary Value Problems via Green-CAS Wavelet Method

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1164 ◽  
Author(s):  
Muhammad Ismail ◽  
Umer Saeed ◽  
Jehad Alzabut ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equations ◽  
Boundary Value ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Wavelet Method ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Cas Wavelets

In this study, we present a novel numerical scheme for the approximate solutions of linear as well as non-linear ordinary differential equations of fractional order with boundary conditions. This method combines Cosine and Sine (CAS) wavelets together with Green function, called Green-CAS method. The method simplifies the existing CAS wavelet method and does not require conventional operational matrices of integration for certain cases. Quasilinearization technique is used to transform non-linear fractional differential equations to linear equations and then Green-CAS method is applied. Furthermore, the proposed method has also been analyzed for convergence, particularly in the context of error analysis. Sufficient conditions for the existence of unique solutions are established for the boundary value problem under consideration. Moreover, to elaborate the effectiveness and accuracy of the proposed method, results of essential numerical applications have also been documented in graphical as well as tabular form.

scholarly journals On non-homogeneous canonical third-order linear differential equations

1994 ◽  
Vol 57 (2) ◽  
pp. 138-148 ◽  
Author(s):  
N. Parhi
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Linear Differential Equations ◽  
Third Order ◽  
Order Linear ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

AbstractIn this paper sufficient conditions have been obtained for non-oscillation of non-homogeneous canonical linear differential equations of third order. Some of these results have been extended to non-linear equations.

scholarly journals A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 275-283
Author(s):  
Kubra Bicer ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Mean Value ◽  
Computational Method ◽  
Mean Value Theorem ◽  
Residual Functions ◽  
Non Linear ◽  
Linear Differential

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

On non-oscillation for certain system of non-linear ordinary differential equations

2017 ◽  
Vol 24 (2) ◽  
pp. 277-285 ◽  
Author(s):  
Zdeněk Opluštil
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Dimensional System ◽  
Two Dimensional ◽  
Linear Ordinary Differential Equations ◽  
Second Order Differential Equations ◽  
Integrable Functions ◽  
Non Linear ◽  
Two Dimensional System ◽  
Non Linear Equations

AbstractWe consider the following two-dimensional system of non-linear equations:u^{\prime}=g(t)|v|^{\frac{1}{\alpha}}\operatorname{sgn}v,\quad v^{\prime}=-p(t% )|u|^{\alpha}\operatorname{sgn}u,where {\alpha>0}, and {g\colon{[0,+\infty[}\rightarrow{[0,+\infty[}} and {p\colon{[0,+\infty[}\rightarrow\mathbb{R}} are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on {[0,+\infty]}. We establish new non-oscillation criteria for the considered system, which generalize known results for the corresponding linear system and for second order differential equations. In particular, the presented criteria are in compliance with the results of Hille and Nehari.

MHD STAGNATION POINT FLOW OF HYPERBOLIC TANGENT FLUID WITH VISCOUS DISSIPATION AND CHEMICAL REACTION

2021 ◽  
Vol 21 (2) ◽  
pp. 569-588
Author(s):  
KINZA ARSHAD ◽  
MUHAMMAD ASHRAF
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stagnation Point ◽  
Viscous Dissipation ◽  
Chemical Reaction ◽  
Normal Velocity ◽  
Hyperbolic Tangent ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

In the present work, two dimensional flow of a hyperbolic tangent fluid with chemical reaction and viscous dissipation near a stagnation point is discussed numerically. The analysis is performed in the presence of magnetic field. The governing partial differential equations are converted into non-linear ordinary differential equations by using appropriate transformation. The resulting higher order non-linear ordinary differential equations are discretized by finite difference method and then solved by SOR (Successive over Relaxation parameter) method. The impact of the relevant parameters is scrutinized by plotting graphs and discussed in details. The main conclusion is that the large value of magnetic field parameter and wiessenberg numbers decrease the streamwise and normal velocity while increase the temperature distribution. Also higher value of the Eckert number Ec results in increases in temperature profile.

Convective transport of thermal and solutal energy in unsteady MHD Oldroyd-B nanofluid flow

2021 ◽  
Author(s):  
Muhammad Yasir ◽  
Masood Khan ◽  
Awais Ahmed ◽  
Malik Zaka Ullah
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Ordinary Differential Equations ◽  
Heat Generation ◽  
Axisymmetric Flow ◽  
Convective Transport ◽  
Buongiorno’S Model ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

Abstract In this work, an analysis is presented for the unsteady axisymmetric flow of Oldroyd-B nanofluid generated by an impermeable stretching cylinder with heat and mass transport under the influence of heat generation/absorption, thermal radiation and first-order chemical reaction. Additionally, thermal and solutal performances of nanofluid are studied using an interpretation of the well-known Buongiorno's model, which helps us to determine the attractive characteristics of Brownian motion and thermophoretic diffusion. Firstly, the governing unsteady boundary layer equation's (PDEs) are established and then converted into highly non-linear ordinary differential equations (ODEs) by using the suitable similarity transformations. For the governing non-linear ordinary differential equations, numerical integration in domain [0, ∞) is carried out using the BVP Midrich scheme in Maple software. For the velocity, temperature and concentration distributions, reliable results are prepared for different physical flow constraints. According to the results, for increasing values of Deborah numbers, the temperature and concentration distribution are higher in terms of relaxation time while these are decline in terms of retardation time. Moreover, thermal radiation and heat generation/absorption are increased the temperature distribution and corresponding boundary layer thickness. With previously stated numerical values, the acquired solutions have an excellent accuracy.

An elliptic boundary value problem for strongly non-linear equations in unbounded domains

1977 ◽  
Vol 77 (3-4) ◽  
pp. 217-230 ◽  
Author(s):  
Vesa Mustonen
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Equations ◽  
Unbounded Domains ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Non Linear ◽  
Linear Elliptic Boundary ◽  
Non Linear Equations

SynopsisThe existence of a variational solution is shown for the strongly non-linear elliptic boundary value problem in unbounded domains. The proof is a generalisation to Orlicz-Sobolev space setting of the idea introduced in [15] for the equations involving polynomial non-linearities only.

scholarly journals On periodic-type solutions of systems of linear ordinary differential equations

2004 ◽  
Vol 2004 (5) ◽  
pp. 395-406 ◽  
Author(s):  
I. Kiguradze
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Type Solution ◽  
Linear Ordinary Differential Equations

We establish nonimprovable, in a certain sense, sufficient conditions for the existence of a unique periodic-type solution for systems of linear ordinary differential equations.

scholarly journals Boundedness of Solutions for Second Order Differential Equations

2020 ◽  
Vol 12 (4) ◽  
pp. 58
Author(s):  
Daniel C. Biles
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Boundedness Of Solutions ◽  
Second Order Differential Equations ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

We present new theorems which specify sufficient conditions for the boundedness of all solutions for second order non-linear differential equations. Unboundedness of solutions is also considered.

scholarly journals On a class of non-linear differential equations with exact solution

2012 ◽  
Vol 34 (1) ◽  
pp. 7-17
Author(s):  
Dao Huy Bich ◽  
Nguyen Dang Bich
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Solid Mechanics ◽  
Linear Equations ◽  
Linear Differential Equations ◽  
Second Order Differential Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

The present paper deals with a class of non-linear ordinary second-order differential equations with exact solutions. A procedure for finding the general exact solution based on a known particular one is derived. For illustration solutions of some non-linear equations occurred in many problems of solid mechanics are considered.

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