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 third order ordinary differential equations directly using hybrid numerical models

2020 ◽  
pp. 69-76
Author(s):  
J. O. Kuboye ◽  
O. F. Quadri ◽  
O. R. Elusakin
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Numerical Models ◽  
Gaussian Elimination ◽  
Initial Value ◽  
Third Order ◽  
Block Methods ◽  
Grid Points

In this work, numerical methods for solving third order initial value problems of ordinary differential equations are developed. Multi-step collocation is used in deriving the methods, where power series approximate solution is employed as a basis function. Gaussian elimination approach is considered in finding the unknown variables $a_j, j=0(1)8$ in interpolation and collocation equations, which are substituted into the power series to give the continuous implicit schemes. The discrete schemes and its derivatives are derived by evaluating the grid and non-grid points. These schemes are arranged in a matrix form to produce block methods. The order of the developed methods are found to be six. The numerical results proved the efficiency of the methods over the existing methods.

scholarly journals A Seven-Step Block Multistep Method for the Solution of First Order Stiff Differential Equations

2020 ◽  
Vol 12 (1) ◽  
pp. 72-82
Author(s):  
Solomon Gebregiorgis ◽  
Hailu Muleta
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Power Series ◽  
Differential Equations ◽  
Multistep Method ◽  
Stiff Differential Equations ◽  
Block Method ◽  
Initial Value ◽  
Numerical Examples ◽  
First Order

In this paper, a seven-step block method for the solution of first order initial value problem in ordinary differential equations based on collocation of the differential equation and interpolation of the approximate solution using power series have been formed. The method is found to be consistent and zero-stable which guarantees convergence. Finally, numerical examples are presented to illustrate the accuracy and effectiveness of the method.  Keywords: Power series, Collocation, Interpolation, Block method, Stiff.

scholarly journals Solutions of first level of meromorphic differential equations

1982 ◽  
Vol 25 (2) ◽  
pp. 183-207 ◽  
Author(s):  
W. Balser
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Convergent Series ◽  
Constant Matrix ◽  
Image Position ◽  
Formal Power ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

Let a meromorphic differential equationbe given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expressionwhere F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, andis a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.

scholarly journals Matrix-valued Bratu equation and the exact solution of its initial value problem

2020 ◽  
pp. 1950007
Author(s):  
Hiroto Inoue
Keyword(s):  
Exact Solution ◽  
Power Series ◽  
Initial Value Problem ◽  
Series Solution ◽  
Power Series Solution ◽  
Matrix Solution ◽  
Initial Value ◽  
Bratu Equation ◽  
Exponential Matrix

A matrix-valued extension of the Bratu equation is defined. For its initial value problem, the exponential matrix solution and power series solution are provided.

scholarly journals Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Nemat Dalir
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Differential Operators ◽  
Nonlinear Partial Differential Equations ◽  
Initial Value ◽  
Third Order ◽  
Partial Differential ◽  
The Third

The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs.

scholarly journals Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

2021 ◽  
Vol 2 (2) ◽  
pp. 62-77
Author(s):  
Rajeev Kumar ◽  
Sanjeev Kumar ◽  
Sukhneet Kaur ◽  
Shrishty Jain
Keyword(s):  
Differential Equations ◽  
Series Solution ◽  
Vries Equation ◽  
Convergent Series ◽  
De Vries ◽  
Conformable Derivatives ◽  
Graphical View ◽  
Conditionally Convergent Series ◽  
The Given ◽  
Explicit Series Solutions

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

scholarly journals A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

2021 ◽  
Vol 9 ◽  
Author(s):  
Ahmad El-Ajou ◽  
Zeyad Al-Zhour
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Hyperbolic System ◽  
Series Solution ◽  
Fractional Power ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

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