Numerical solution of bipolar fuzzy initial value problem

2021 ◽  
Vol 40 (1) ◽  
pp. 1309-1341
Author(s):  
Muhammad Saqib ◽  
Muhammad Akram ◽  
Shahida Bashir ◽  
Tofigh Allahviranloo
Keyword(s):  
Differential Equations ◽  
Fuzzy Set ◽  
Taylor Expansion ◽  
Euler Method ◽  
Mathematical Tool ◽  
Initial Value ◽  
Step Size ◽  
Fuzzy Environment ◽  
Different Types ◽  
Natural Way

Differential equations occur in many fields of science, engineering and social science as it is a natural way of modeling uncertain dynamical systems. A bipolar fuzzy set model is useful mathematical tool for addressing uncertainty which is an extension of fuzzy set model. In this paper, we study differential equations in bipolar fuzzy environment. We introduce the concept gH-derivative of bipolar fuzzy valued function. We present some properties of gH-differentiability of bipolar fuzzy valued function by considering different types of differentiability. We consider bipolar fuzzy Taylor expansion. By using Taylor expansion, Euler method is presented for solving bipolar fuzzy initial value problems. We discuss convergence analysis of proposed method. We describe some numerical examples to see the convergence and stability of the method and compute global truncation error. From numerical results, we see that for small step size Euler method converges to exact solution.

scholarly journals A Quantitative Comparison of Numerical Method for Solving Stiff Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computation Time ◽  
Quantitative Comparison ◽  
Initial Value ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Variable Step

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

scholarly journals Solving Fuzzy Differential Equations by Using Power Series

2020 ◽  
pp. 92-107
Author(s):  
Rasha H. Ibraheem
Keyword(s):  
Approximate Solution ◽  
Power Series ◽  
Differential Equations ◽  
Taylor Expansion ◽  
Series Solution ◽  
Convergent Series ◽  
Initial Value ◽  
Third Order ◽  
Technical Accuracy ◽  
Made In

In this paper, the series solution is applied to solve third order fuzzy differential equations with a fuzzy initial value. The proposed method applies Taylor expansion in solving the system and the approximate solution of the problem which is calculated in the form of a rapid convergent series; some definitions and theorems are reviewed as a basis in solving fuzzy differential equations. An example is applied to illustrate the proposed technical accuracy. Also, a comparison between the obtained results is made, in addition to the application of the crisp solution, when the-level equals one.

scholarly journals Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
F. F. Ngwane ◽  
S. N. Jator
Keyword(s):  
Differential Equations ◽  
Stability Property ◽  
Telegraph Equation ◽  
Hyperbolic Partial Differential Equations ◽  
Initial Value ◽  
Step Size ◽  
Grid Points ◽  
The Stability ◽  
Predictor Corrector ◽  
Trigonometrically Fitted Method

We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

scholarly journals T-stability of the Euler method for impulsive stochastic differential equations driven by fractional Brownian motion

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6493-6503
Author(s):  
Hui Yu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Euler Method ◽  
Variable Step Size ◽  
Step Size ◽  
Theoretical Results

Due to the fact that a fractional Brownian motion (fBm) with the Hurst parameter H ? (0,1/2) U(1/2, 1) is neither a semimartingale nor a Markov process, relatively little is studied about the T-stability for impulsive stochastic differential equations (ISDEs) with fBm. Here, for such linear equations with H ? (1/3, 1/2), by means of the average stability function, sufficient conditions of the T-stability are presented to their numerical solutionswhich are established fromthe Euler-Maruyama method with variable step-size. Moreover, some numerical examples are presented to support the theoretical results.

Fuzzy form of Euler Method to Solve Fuzzy Differential Equations

2021 ◽  
Vol 01 ◽  
Author(s):  
Umme Salma Pirzada ◽  
S. Rama Mohan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riccati Equation ◽  
Initial Value Problems ◽  
Euler Method ◽  
Fuzzy Differential Equation ◽  
Fuzzy Arithmetic ◽  
Local Error ◽  
Initial Value

: This paper proposes fuzzy form of Euler method to solve fuzzy initial value problems. By this method, fuzzy differential equations can be solved directly using fuzzy arithmetic. The solution by this method is readily available in a form of fuzzy-valued function. The method does not require to re-write fuzzy differential equation into system of two crisp ordinary differential equations. Algorithm of the method and local error expression are discussed. An illustration and solution of fuzzy Riccati equation are provided for the applicability of the method.

scholarly journals Comparison of Numerical Methods for Solving Initial Value Problems for Stiff Differential Equations

1970 ◽  
Vol 30 ◽  
pp. 122-132
Author(s):  
Sharaban Thohura ◽  
Azad Rahman
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Procedure ◽  
Stiff Systems ◽  
Stiff Differential Equations ◽  
Initial Value ◽  
Step Size ◽  
Stepsize Control ◽  
Adaptive Stepsize ◽  
Runge Kutta

Special classes of Initial value problem of differential equations termed as stiff differential equations occur naturally in a wide variety of applications including the studies of spring and damping systems, chemical kinetics, electrical circuits, and so on. Most realistic stiff systems do not have analytical solutions so that a numerical procedure must be used. In this paper we have discussed the phenomenon of stiffness and the general purpose procedures for the solution of stiff differential equation. Because of their applications in many branches of engineering and science, many algorithms have been proposed to solve such problems. In this study we have focused on some conventional methods namely Runge-Kutta method, Adaptive Stepsize Control for Runge-Kutta and an ODE Solver package, EPISODE. We describe the characteristics shared by these methods. We compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the traditional numerical methods such as Euler, explicit Runge-Kutta and Adams –Moulton methods step size need to be very small. This however introduces enough round-off errors to cause instability of the solution. To overcome this problem we have used two other algorithms namely Adaptive Stepsize Control for Runge-Kutta and EPISODE. The results are compared with exact one to determine the efficiency of the above mentioned method. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 121-132  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8509

scholarly journals Solving Ordinary Differential Equation of Higher Order by Adomian Decomposition Method

2020 ◽  
pp. 44-53
Author(s):  
Sumayah Ghaleb Othman ◽  
Yahya Qaid Hasan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Higher Order ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Different Types

Aims/ Objectives: In this article, we use Adomian Decomposition method (ADM) for solving initial value problems in the higher order ordinary differential equations. Many researchers have used the ADM in order to find convergent as well as exact solutions of different types of equations. Therefore, the ADM is considered as an effective and successful method for solving differential equations. In this paper, we presented some suggested amendments to the ADM by using a new differential operator in order to find solutions for higher order types of equations. We demonstrated the effectiveness of this method through many examples and we find out that we get an approximate solutions using the proposed amendments. We can conclude that the suggested modification of ADM is afftective and produces reliable results.

scholarly journals Accuracy Analysis on Solution of Initial Value Problems of Ordinary Differential Equations for Some Numerical Methods with Different Step Sizes

2021 ◽  
Vol 10 (1) ◽  
pp. 118-133
Author(s):  
Mohammad Asif Arefin ◽  
Biswajit Gain ◽  
Rezaul Karim
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Numerical Solutions ◽  
Initial Value ◽  
Step Size ◽  
Runge Kutta Method ◽  
Efficiency Comparison

In this article, three numerical methods namely Euler’s, Modified Euler, and Runge-Kutta method have been discussed, to solve the initial value problem of ordinary differential equations. The main goal of this research paper is to find out the accurate results of the initial value problem (IVP) of ordinary differential equations (ODE) by applying the proposed methods. To achieve this goal, solutions of some IVPs of ODEs have been done with the different step sizes by using the proposed three methods, and solutions for each step size are analyzed very sharply. To ensure the accuracy of the proposed methods and to determine the accurate results, numerical solutions are compared with the exact solutions. It is observed that numerical solutions are best fitted with exact solutions when the taken step size is very much small. Consequently, all the proposed three methods are quite efficient and accurate for solving the IVPs of ODEs. Error estimation plays a significant role in the establishment of a comparison among the proposed three methods. On the subject of accuracy and efficiency, comparison is successfully implemented among the proposed three methods.

scholarly journals Results on initial value problems for random fuzzy fractional functional differential equations

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2601-2624
Author(s):  
Ho Vu ◽  
Ngo Hoa ◽  
Nguyen Son ◽  
Donal O’Regane
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Euler Method ◽  
Successive Approximations ◽  
Initial Value ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Functional Differential ◽  
Fractional Functional Differential Equations ◽  
Fractional Functional

In this paper random fuzzy fractional functional differential equations (RFFFDEs) with Caputo generalized Hukuhara differentiability are introduced. We present existence and uniqueness results for RFFFDEs using the idea of successive approximations. The behaviour of solutions when the data of the equation are subjected to errors is discussed. Furthermore, the solution to random fuzzy fractional functional initial value problem under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method (MFEM) is presented. The results are illustrated with examples.

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