Method of Separation of Variables for the Solution of Certain Nonlinear Partial Differential Equations

1971 ◽  
Vol 93 (2) ◽  
pp. 162-164
Author(s):  
V. A. Bapat ◽  
P. Srinivasan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Separation Of Variables ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Exact Methods ◽  
Partial Differential ◽  
Approximate Methods

A method for the solution of a certain class of nonlinear partial differential equations by the method of separation of variables is presented. The method enables the nonlinear partial differential equation to be reduced to ordinary nonlinear differential equations, which can be solved by exact methods (or by approximate methods if an exact solution is not possible).

scholarly journals Transient-False Method for Solving System of Nonlinear Partial Differential Equations

2018 ◽  
Vol 1 (25) ◽  
pp. 509-522
Author(s):  
. Ali Khalaf Hussain
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations

          In this paper we study the false transient method  to  solve and transform a system of non-linear partial differential equations which can be solved using finite-difference method and give some problems which have a good results compared with the exact solution, whereas this method was used to transform the nonlinear partial differential equation to a linear partial differential equation which can be solved by using the alternating-direction implicit method after using the ADI method. The system of linear algebraic equations could be obtained and can be solved by using MATLAB.

scholarly journals Modeling Nonlinear Partial Differential Equations and Construction of Solitary Wave Solutions in an Inductive Electrical Line

2019 ◽  
pp. 1-10 ◽  
Author(s):  
Tiague Takongmo Guy ◽  
Jean Roger Bogning
Keyword(s):  
Partial Differential Equations ◽  
Solitary Wave ◽  
Magnetic Flux ◽  
Differential Equations ◽  
Solitary Waves ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Hyperbolic Function ◽  
Flux Linkage ◽  
Partial Differential

A soliton is considered nowadays as a future wave reason being the fact that it is a stable, robust and non-dissipative solitary wave. If one uses a soliton as a transmission signal in electrical lines, this will have a great impacts in the domain of economic, technology and education. Given the fact that the propagation of the soliton is due to the interaction between dispersion and nonlinearity, it necessitates that the transmission medium should be dispersive and nonlinear. The physical system we have chosen for our survey is an inductive electrical line reason being the fact that it is the cheapest and very easy to manufacture than any other transmission lines; furthermore we find out the analytical variation that the magnetic flux linkage of inductors in the electrical line must undergo so that its transmission medium admits the propagation of solitary waves of required type. The aim of this work is to model nonlinear partial differential equations which govern the dynamics of those solitary waves in the line, to define the analytical expression of the magnetic flux linkage of inductors in the line and to find out some exact solutions of solitary waves types of those equations. To meet our objectives, we apply Kirchhoff laws to the circuit of a nonlinear inductive electrical line to model the nonlinear partial differential equation which describe the dynamics of those solitons. Further we apply the effective and direct Bogning-Djeumen Tchaho-Kofane method based on the identification of basic hyperbolic function coefficients to construct some exact soliton solutions of modeled equations. Numerical simulations have enabled to draw and observe the real profile of those solitary waves which are Kink soliton and Pulse soliton. The obtained results are supposed to permits: The facilitation of the choice of the type of line relative to the type of signal one wishes to send across, to increase the mathematical field knowledge, the reduction of amplification stations of those lines, The manufacturing of new inductors and new electrical lines susceptible of propagating those solitary waves.

scholarly journals Parametric Analysis of Entropy Generation in Off-centered Stagnation Flow Towards a Rotating Disc

2014 ◽  
Vol 3 (1) ◽  
pp. 27-41 ◽  
Author(s):  
M.M. Rashidi ◽  
L. Shamekhi ◽  
Sunil Kumar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Entropy Generation ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Stagnation Flow ◽  
Rotating Disc ◽  
Partial Differential ◽  
Keller Box Method ◽  
Box Method

AbstractThe similarity solution for the steady stagnation flow towards an off-centered rotating disc is gives a systemof non-linear partial differential equations. These nonlinear differential equations are numerically solved by applyingwell known Keller-Box Method. After finding the velocity distributions, the important designing subject, entropy generation of this system has been analyzed. Graphical results are presented to investigate effects of the rotation ratio α, off-centering, Reynolds number and axial height on the radial and azimuthal velocities and entropy generation. In order to show the effectiveness of the Keller- Box method, the obtained results are compared with available solutions obtained using DTM. The obtained results demonstrate the reliability of the algorithm and the Keller- Box method is an attractive method in solving the systems of nonlinear partial differential equations, and also the entropy generation is an important parameter depends on design andwork conditions that should be in the attention of designers of these rotating systems.

Numerical Reservoir Simulation Using an Ordinary-Differential-Equations Integrator

1975 ◽  
Vol 15 (03) ◽  
pp. 255-264 ◽  
Author(s):  
R.F. Sincovec
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Reservoir Simulation ◽  
Nonlinear Partial Differential Equations ◽  
Method Of Lines ◽  
Variable Order ◽  
Partial Differential

Abstract The method of lines used in conjunction with a sophisticated ordinary-differential-equations integrator is an effective approach for solving nonlinear, partial differential equations and is applicable to the equations describing fluid flow through porous media. Given the initial values, the integrator is self-starting. Subsequently, it automatically and reliably selects the time step and order, solves the nonlinear equations (checking for convergence, etc.), and maintains a user-specified time-integration accuracy, while attempting to complete the problems in a minimal amount of computer time. The advantages of this approach, such as stability, accuracy, reliability, and flexibility, are discussed. The method is applied to reservoir simulation, including high-rate and gas-percolation problems, and appears to be readily applicable to problems, and appears to be readily applicable to compositional models. Introduction The numerical solution of nonlinear, partial differential equations is usually a complicated and lengthy problem-dependent process. Generally, the solution of slightly different types of partial differential equations requires an entirely different computer program. This situation for partial differential equations is in direct contrast to that for ordinary differential equations. Recently, sophisticated and highly reliable computer programs for automatically solving complicated systems of ordinary differential equations have become available. These computer programs feature variable-order methods and automatic time-step and error control, and are capable of solving broad classes of ordinary differential equations. This paper discusses how these sophisticated ordinary-differential-equation integrators may be used to solve systems of nonlinear partial differential equations. partial differential equations.The basis for the technique is the method of lines. Given a system of time-dependent partial differential equations, the spatial variable(s) are discretized in some manner. This procedure yields an approximating system of ordinary differential equations that can be numerically integrated with one of the recently developed, robust ordinary-differential-equation integrators to obtain numerical approximations to the solution of the original partial differential equations. This approach is not new, but the advent of robust ordinary-differential-equation integrators has made the numerical method of lines a practical and efficient method of solving many difficult systems of partial differential equations. The approach can be viewed as a variable order in time, fixed order in space technique. Certain aspects of this approach are discussed and advantages over more conventional methods are indicated. Use of ordinary-differential-equation integrators for simplifying the heretofore rather complicated procedures for accurate numerical integration of systems of nonlinear, partial differential equations is described. This approach is capable of eliminating much of the duplicate programming effort usually associated with changing equations, boundary conditions, or discretization techniques. The approach can be used for reservoir simulation, and it appears that a compositional reservoir simulator can be developed with relative ease using this approach. In particular, it should be possible to add components to or delete components possible to add components to or delete components from the compositional code with only minor modifications. SPEJ P. 255

THE MODIFIED DIFFERENTIAL TRANSFORM METHOD FOR SOLVING OFF-CENTERED STAGNATION FLOW TOWARD A ROTATING DISC

2010 ◽  
Vol 07 (04) ◽  
pp. 655-670 ◽  
Author(s):  
ESMAEEL ERFANI ◽  
MOHAMMAD MEHDI RASHIDI ◽  
AMIR BASIRI PARSA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Stagnation Flow ◽  
Rotating Disc ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform

The similarity solution for the steady stagnation flow toward an off-centered rotating disc gives a system of nonlinear partial differential equations. These nonlinear differential equations are analytically solved by applying a newly developed method called DTM–Padé technique (the combination of the differential transform method (DTM) and the Padé approximation). This technique is extended to give solutions for nonlinear differential equations with boundary conditions at infinity. Graphical results are presented to investigate influence of the rotation ratio α on the radial velocity, azimuthal velocity, and the induced velocity. In order to show the effectiveness of the DTM–Padé technique, the results obtained from the DTM–Padé technique are compared with available solutions obtained using shooting method to generate the numerical solution. The obtained results demonstrate the reliability of the algorithm and the DTM–Padé technique is an attractive method in solving the systems of nonlinear partial differential equations.

scholarly journals A REVIEW ON NUMERICAL METHODS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 23 (07) ◽  
pp. 1342-1352
Author(s):  
◽  
Dr. Jatinder kaur ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Wave Equations ◽  
Numerical Models ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Non Linear ◽  
Linear Differential

Progression in innovation and engineering presents us with numerous difficulties, comparably to conquer such engineering difficulties with the assistance of various numerical models, equations are taken. Since in the first place Mathematicians, Designers and Engineers make progress toward accuracy what’s more, exactness while addressing equations Differential equations, specifically, hold an enormous application in engineering and numerous different areas. One such sort of Differential equation is known as partial differential equation. The range of application of partial differential equations comprises of recreation, calculation age, and investigation of higher request PDE and wave equations. Adjusting diverse numerical methods prompts an assortment of answers and contrast among them, subsequently the determination of the method of addressing is one of the urgent boundaries to produce exact outcomes. Our work centres’ around the survey of various numerical methods to settle Non-linear differential equations based on exactness and effectiveness, in order to diminish the emphases. These would orchestrate rules to existing numerical methods of nonlinear partial differential equations.[1]

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