scholarly journals Construction of Benchmark Problems for Solution of Ordinary Differential Equations

1994 ◽  
Vol 1 (5) ◽  
pp. 403-414 ◽  
Author(s):  
Sangchul Lee ◽  
John L. Junkins
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Solution ◽  
Solution Process ◽  
Optimal Choice ◽  
Benchmark Problems ◽  
Tuning Parameters ◽  
Forcing Function ◽  
The Relationship

An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed in this fashion have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially available approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Five illustrative examples are presented.

scholarly journals Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
S. Narayanamoorthy ◽  
T. L. Yookesh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Approximate Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Delay Differential

We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

scholarly journals Numerical solution of nonlinear system of integro-differential equations using Chebyshev wavelets

2015 ◽  
Vol 11 (2) ◽  
pp. 15-34
Author(s):  
H. Aminikhah ◽  
S. Hosseini
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Wavelets ◽  
Linear And Nonlinear

Abstract This paper introduces an approach for obtaining the numerical solution of the linear and nonlinear integro-differential equations using Chebyshev wavelets approximations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results have been compared with the exact solution. Comparison of the approximate solution with exact solution shows that the used method is effectiveness and practical for classes of linear and nonlinear system of integro-differential equations.

Numerical Solution for a System of Fractional Differential Equations with Applications in Fluid Dynamics and Chemical Engineering

2017 ◽  
Vol 15 (5) ◽  
Author(s):  
Bijil Prakash ◽  
Amit Setia ◽  
Shourya Bose
Keyword(s):  
Fluid Dynamics ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Crucial Role ◽  
Chemical Engineering ◽  
Haar Wavelets ◽  
Fractional Differential

Abstract In this paper, a Haar wavelets based numerical method to solve a system of linear or nonlinear fractional differential equations has been proposed. Numerous nontrivial test examples along with practical problems from fluid dynamics and chemical engineering have been considered to illustrate applicability of the proposed method. We have derived a theoretical error bound which plays a crucial role whenever the exact solution of the system is not known and also it guarantees the convergence of approximate solution to exact solution.

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.

A family of positivity preserving schemes for numerical solution of Black–Scholes equation

2016 ◽  
Vol 03 (04) ◽  
pp. 1650025
Author(s):  
M. Mehdizadeh Khalsaraei ◽  
R. Shokri Jahandizi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Numerical Solution ◽  
Numerical Scheme ◽  
Initial Vector ◽  
Positivity Preserving ◽  
Fully Implicit ◽  
Black Scholes ◽  
Spurious Oscillations

When one solves the Black–Scholes partial differential equation, it is of great important that numerical scheme to be free of spurious oscillations and satisfy the positivity requirement. With positivity, we mean, the component non-negativity of the initial vector, is preserved in time for the exact solution. Numerically, such property for fully implicit scheme is not always satisfied by approximated solutions and they generate spurious oscillations in the presence of discontinuous payoff. In this paper, by using the nonstandard discretization strategy, we propose a new scheme that is free of spurious oscillations and satisfies the positivity requirement.

scholarly journals THE NUMERICAL SOLUTION BY THE METHOD OF DIRECT INTEGRALS OF DIFFERENTIATION OF EQUATIONS HAVE AN APPLICATION IN THE GAS FILTRATION THEOREM

2020 ◽  
pp. 147-153
Author(s):  
Abdujabbor Abdurazakov ◽  
Nasiba Makhmudova ◽  
Nilufar Mirzamakhmudova
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Error Estimate ◽  
Numerical Solution ◽  
Direct Method ◽  
Gas Filtration ◽  
Sweep Method ◽  
Time Step

On the basis of the direct method and a combination of differential sweep, the article developed a calculated algorithm for solving gas filtration, thereby taking into account the convergence of the approximate solution to the exact one. KEYWORDS: direct method, sweep method, differential equation, time step, convergence, approximate solution, error estimate.

scholarly journals Numerical Solution Of Partial Integro-Differential Equation Using Legendre Multi Wavelets

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Amina Kassim Hussain
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Physical Phenomenon ◽  
Diffusion Method ◽  
Benchmark Problems ◽  
Science And Engineering ◽  
Partial Differential ◽  
Integro Differential Equation

It is very important to state the initial assumptions for the description of physical phenomenon in the case of partial integro-differential equation. The parabolic equations and boundary conditions can be used to define the time-dependent diffusion process. The integro-differential equation is the combination of integration and derivatives. It is part of the technology, which includes science and engineering. Various models that cover the area of science and engineering are available. Moreover, variable techniques are accessible to solve the integro-differential equations. Numerical method is an important way to solve the challenges in the field of science and industry. To improve efficiency, the companies were working on computer simulation. For reliability, flexibility, and inexpensiveness, the numerical methods are preferred. Linear Legendre multi wavelets form a collocated method based on the numerical solution of one-dimensional parabolic partial integro-differential equation of diffusion type. In this study, we aim to study the diffusion method of numerical solution for the integro-partial differential equation. The diffusion method, its basic concept, and other methods used to solve integro-partial differential equations are also studied in detail. The proposed numerical method is useful to different benchmark problems and provides efficient, accurate, and robust results.

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