**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**2533

# Search results for: Fuzzy differential equation

##### 2533 Simulink Approach to Solve Fuzzy Differential Equation under Generalized Differentiability

**Authors:**
N. Kumaresan ,
J. Kavikumar,
Kuru Ratnavelu

**Abstract:**

**Keywords:**
Fuzzy differential equation,
Generalized differentiability,
H-difference and Simulink.

##### 2532 Solution of Fuzzy Differential Equation under Generalized Differentiability by Genetic Programming

**Authors:**
N. Kumaresan,
J. Kavikumar,
M. Kumudthaa,
Kuru Ratnavelu

**Abstract:**

**Keywords:**
Fuzzy differential equation,
Generalized differentiability,
Genetic programming and H-difference.

##### 2531 Adomian Method for Second-order Fuzzy Differential Equation

**Authors:**
Lei Wang,
Sizong Guo

**Abstract:**

In this paper, we study the numerical method for solving second-order fuzzy differential equations using Adomian method under strongly generalized differentiability. And, we present an example with initial condition having four different solutions to illustrate the efficiency of the proposed method under strongly generalized differentiability.

**Keywords:**
Fuzzy-valued function,
fuzzy initial value problem,
strongly generalized differentiability,
adomian decomposition method.

##### 2530 Strict Stability of Fuzzy Differential Equations by Lyapunov Functions

**Authors:**
Mustafa Bayram Gücen,
Coşkun Yakar

**Abstract:**

**Keywords:**
Fuzzy systems,
fuzzy differential equations,
fuzzy
stability,
strict stability.

##### 2529 Complex Fuzzy Evolution Equation with Nonlocal Conditions

**Authors:**
Abdelati El Allaoui,
Said Melliani,
Lalla Saadia Chadli

**Abstract:**

**Keywords:**
Complex fuzzy evolution equations,
nonlocal
conditions,
mild solution,
complex fuzzy semigroups.

##### 2528 Strict Stability of Fuzzy Differential Equations with Impulse Effect

**Authors:**
Sanjay K.Srivastava,
Bhanu Gupta

**Abstract:**

In this paper some results on strict stability heve beeb extended for fuzzy differential equations with impulse effect using Lyapunov functions and Razumikhin technique.

**Keywords:**
Fuzzy differential equations,
Impulsive differential equations,
Strict stability,
Lyapunov function,
Razumikhin technique.

##### 2527 Existence of Iterative Cauchy Fractional Differential Equation

**Authors:**
Rabha W. Ibrahim

**Abstract:**

Our main aim in this paper is to use the technique of non expansive operators to more general iterative and non iterative fractional differential equations (Cauchy type ). The non integer case is taken in sense of Riemann-Liouville fractional operators. Applications are illustrated.

**Keywords:**
Fractional calculus,
fractional differential equation,
Cauchy equation,
Riemann-Liouville fractional operators,
Volterra
integral equation,
non-expansive mapping,
iterative differential equation.

##### 2526 On the Fuzzy Difference Equation xn+1 = A +

**Authors:**
Qianhong Zhang,
Lihui Yang,
Daixi Liao,

**Abstract:**

In this paper, we study the existence, the boundedness and the asymptotic behavior of the positive solutions of a fuzzy nonlinear difference equations xn+1 = A + k i=0 Bi xn-i , n= 0, 1, · · · . where (xn) is a sequence of positive fuzzy numbers, A,Bi and the initial values x-k, x-k+1, · · · , x0 are positive fuzzy numbers. k ∈ {0, 1, 2, · · ·}.

**Keywords:**
Fuzzy difference equation,
boundedness,
persistence,
equilibrium point,
asymptotic behaviour.

##### 2525 Numerical Study of Some Coupled PDEs by using Differential Transformation Method

**Authors:**
Reza Abazari,
Rasool Abazari

**Abstract:**

In this paper, the two-dimension differential transformation method (DTM) is employed to obtain the closed form solutions of the three famous coupled partial differential equation with physical interest namely, the coupled Korteweg-de Vries(KdV) equations, the coupled Burgers equations and coupled nonlinear Schrödinger equation. We begin by showing that how the differential transformation method applies to a linear and non-linear part of any PDEs and apply on these coupled PDEs to illustrate the sufficiency of the method for this kind of nonlinear differential equations. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.

**Keywords:**
Coupled Korteweg-de Vries(KdV) equation,
Coupled Burgers equation,
Coupled Schrödinger equation,
differential transformation method.

##### 2524 Adomian Decomposition Method Associated with Boole-s Integration Rule for Goursat Problem

**Authors:**
Mohd Agos Salim Nasir,
Ros Fadilah Deraman,
Siti Salmah Yasiran

**Abstract:**

**Keywords:**
Goursat problem,
partial differential equation,
Adomian decomposition method,
Boole's integration rule.

##### 2523 Step Method for Solving Nonlinear Two Delays Differential Equation in Parkinson’s Disease

**Authors:**
H. N. Agiza,
M. A. Sohaly,
M. A. Elfouly

**Abstract:**

Parkinson's disease (PD) is a heterogeneous disorder with common age of onset, symptoms, and progression levels. In this paper we will solve analytically the PD model as a non-linear delay differential equation using the steps method. The step method transforms a system of delay differential equations (DDEs) into systems of ordinary differential equations (ODEs). On some numerical examples, the analytical solution will be difficult. So we will approximate the analytical solution using Picard method and Taylor method to ODEs*.*

**Keywords:**
Parkinson's disease,
Step method,
delay differential equation,
simulation.

##### 2522 Noise Analysis of Single-Ended Input Differential Amplifier using Stochastic Differential Equation

**Authors:**
Tarun Kumar Rawat,
Abhirup Lahiri,
Ashish Gupta

**Abstract:**

In this paper, we analyze the effect of noise in a single- ended input differential amplifier working at high frequencies. Both extrinsic and intrinsic noise are analyzed using time domain method employing techniques from stochastic calculus. Stochastic differential equations are used to obtain autocorrelation functions of the output noise voltage and other solution statistics like mean and variance. The analysis leads to important design implications and suggests changes in the device parameters for improved noise characteristics of the differential amplifier.

**Keywords:**
Single-ended input differential amplifier,
Noise,
stochastic differential equation,
mean and variance.

##### 2521 An Analytical Method for Solving General Riccati Equation

**Authors:**
Y. Pala,
M. O. Ertas

**Abstract:**

In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

**Keywords:**
Riccati Equation,
ordinary differential equation,
nonlinear differential equation,
analytical solution,
proper solution.

##### 2520 Position Vector of a Partially Null Curve Derived from a Vector Differential Equation

**Authors:**
Süha Yılmaz,
Emin Özyılmaz,
Melih Turgut,
Şuur Nizamoğlu

**Abstract:**

In this paper, position vector of a partially null unit speed curve with respect to standard frame of Minkowski space-time is studied. First, it is proven that position vector of every partially null unit speed curve satisfies a vector differential equation of fourth order. In terms of solution of the differential equation, position vector of a partially null unit speed curve is expressed.

**Keywords:**
Frenet Equations,
Partially Null Curves,
Minkowski Space-time,
Vector Differential Equation.

##### 2519 A Genetic Algorithm Approach for Solving Fuzzy Linear and Quadratic Equations

**Authors:**
M. Hadi Mashinchi,
M. Reza Mashinchi,
Siti Mariyam H. J. Shamsuddin

**Abstract:**

In this paper a genetic algorithms approach for solving the linear and quadratic fuzzy equations Ãx̃=B̃ and Ãx̃^{2} + B̃x̃=C̃ , where Ã, B̃, C̃ and x̃ are fuzzy numbers is proposed by genetic algorithms. Our genetic based method initially starts with a set of random fuzzy solutions. Then in each generation of genetic algorithms, the solution candidates converge more to better fuzzy solution x̃_{b} . In this proposed method the final reached x̃_{b} is not only restricted to fuzzy triangular and it can be fuzzy number.

**Keywords:**
Fuzzy coefficient,
fuzzy equation,
genetic
algorithms.

##### 2518 Instability of a Nonlinear Differential Equation of Fifth Order with Variable Delay

**Authors:**
Cemil Tunc

**Abstract:**

In this paper, we study the instability of the zero solution to a nonlinear differential equation with variable delay. By using the Lyapunov functional approach, some sufficient conditions for instability of the zero solution are obtained.

**Keywords:**
Instability,
Lyapunov-Krasovskii functional,
delay differential equation,
fifth order.

##### 2517 Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations

**Authors:**
Fuziyah Ishak,
Siti Norazura Ahmad

**Abstract:**

Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

**Keywords:**
Accuracy,
extended trapezoidal method,
numerical solution,
Volterra integro-differential equations.

##### 2516 Stability Analysis of Two-delay Differential Equation for Parkinson's Disease Models with Positive Feedback

**Authors:**
M. A. Sohaly,
M. A. Elfouly

**Abstract:**

Parkinson's disease (PD) is a heterogeneous movement disorder that often appears in the elderly. PD is induced by a loss of dopamine secretion. Some drugs increase the secretion of dopamine. In this paper, we will simply study the stability of PD models as a nonlinear delay differential equation. After a period of taking drugs, these act as positive feedback and increase the tremors of patients, and then, the differential equation has positive coefficients and the system is unstable under these conditions. We will present a set of suggested modifications to make the system more compatible with the biodynamic system. When giving a set of numerical examples, this research paper is concerned with the mathematical analysis, and no clinical data have been used.

**Keywords:**
Parkinson's disease,
stability,
simulation,
two delay differential equation.

##### 2515 Group Invariant Solutions of Nonlinear Time-Fractional Hyperbolic Partial Differential Equation

**Authors:**
Anupma Bansal,
Rajeev Budhiraja,
Manoj Pandey

**Abstract:**

**Keywords:**
Nonlinear time-fractional hyperbolic PDE,
Lie
Classical method,
exact solutions.

##### 2514 Stability Analysis of Linear Fractional Order Neutral System with Multiple Delays by Algebraic Approach

**Authors:**
Lianglin Xiong,
Yun Zhao,
Tao Jiang

**Abstract:**

In this paper, we study the stability of n-dimensional linear fractional neutral differential equation with time delays. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. An example is provided to show the effectiveness of the approach presented in this paper.

**Keywords:**
Fractional neutral differential equation,
Laplace transform,
characteristic equation.

##### 2513 Periodic Solutions for a Higher Order Nonlinear Neutral Functional Differential Equation

**Authors:**
Yanling Zhu

**Abstract:**

In this paper, a higher order nonlinear neutral functional differential equation with distributed delay is studied by using the continuation theorem of coincidence degree theory. Some new results on the existence of periodic solutions are obtained.

**Keywords:**
Neutral functional differential equation,
higher order,
periodic solution,
coincidence degree theory.

##### 2512 Stability of Fractional Differential Equation

**Authors:**
Rabha W. Ibrahim

**Abstract:**

We study a Dirichlet boundary value problem for Lane-Emden equation involving two fractional orders. Lane-Emden equation has been widely used to describe a variety of phenomena in physics and astrophysics, including aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres,and thermionic currents. However, ordinary Lane-Emden equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractalmedium, numerous generalizations of Lane-Emden equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Lane-Emden equation. This gives rise to the fractional Lane-Emden equation with a single index. Recently, a new type of Lane-Emden equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskiis fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space. Ulam-Hyers stability for iterative Cauchy fractional differential equation is defined and studied.

**Keywords:**
Fractional calculus,
fractional differential equation,
Lane-Emden equation,
Riemann-Liouville fractional operators,
Volterra integral equation.

##### 2511 Improving Load Frequency Control of Multi-Area Power System by Considering Uncertainty by Using Optimized Type 2 Fuzzy Pid Controller with the Harmony Search Algorithm

**Authors:**
Mehrdad Mahmudizad,
Roya Ahmadi Ahangar

**Abstract:**

**Keywords:**
Load Frequency Control,
Fuzzy-PID controller,
Type 2 fuzzy system,
Harmony Search algorithm.

##### 2510 Existence of Solution for Singular Two-point Boundary Value Problem of Second-order Differential Equation

**Authors:**
Xiguang Li

**Abstract:**

In this paper, by constructing a special set and utilizing fixed point theory in coin, we study the existence of solution of singular two point’s boundary value problem for second-order differential equation, which improved and generalize the result of related paper.

**Keywords:**
Singular differential equation,
boundary value problem,
coin,
fixed point theory.

##### 2509 Explicit Solutions and Stability of Linear Differential Equations with multiple Delays

**Authors:**
Felix Che Shu

**Abstract:**

We give an explicit formula for the general solution of a one dimensional linear delay differential equation with multiple delays, which are integer multiples of the smallest delay. For an equation of this class with two delays, we derive two equations with single delays, whose stability is sufficient for the stability of the equation with two delays. This presents a new approach to the study of the stability of such systems. This approach avoids requirement of the knowledge of the location of the characteristic roots of the equation with multiple delays which are generally more difficult to determine, compared to the location of the characteristic roots of equations with a single delay.

**Keywords:**
Delay Differential Equation,
Explicit Solution,
Exponential
Stability,
Lyapunov Exponents,
Multiple Delays.

##### 2508 Comparison Results of Two-point Fuzzy Boundary Value Problems

**Authors:**
Hsuan-Ku Liu

**Abstract:**

This paper investigates the solutions of two-point fuzzy boundary value problems as the form x = f(t, x(t)), x(0) = A and x(l) = B, where A and B are fuzzy numbers. There are four different solutions for the problems when the lateral type of H-derivative is employed to solve the problems. As f(t, x) is a monotone function of x, these four solutions are reduced to two different solutions. As f(t, x(t)) = λx(t) or f(t, x(t)) = -λx(t), solutions and several comparison results are presented to indicate advantages of each solution.

**Keywords:**
Fuzzy derivative,
lateral type of H-derivative,
fuzzy differential equations,
fuzzy boundary value problems,
boundary value problems.

##### 2507 Unique Positive Solution of Nonlinear Fractional Differential Equation Boundary Value Problem

**Authors:**
Fengxia Zheng

**Abstract:**

By using two new fixed point theorems for mixed monotone operators, the positive solution of nonlinear fractional differential equation boundary value problem is studied. Its existence and uniqueness is proved, and an iterative scheme is constructed to approximate it.

**Keywords:**
Fractional differential equation,
boundary value problem,
positive solution,
existence and uniqueness,
fixed point theorem,
mixed monotone operator.

##### 2506 Exp-Function Method for Finding Some Exact Solutions of Rosenau Kawahara and Rosenau Korteweg-de Vries Equations

**Authors:**
Ehsan Mahdavi

**Abstract:**

In this paper, we apply the Exp-function method to Rosenau-Kawahara and Rosenau-KdV equations. Rosenau-Kawahara equation is the combination of the Rosenau and standard Kawahara equations and Rosenau-KdV equation is the combination of the Rosenau and standard KdV equations. These equations are nonlinear partial differential equations (NPDE) which play an important role in mathematical physics. Exp-function method is easy, succinct and powerful to implement to nonlinear partial differential equations arising in mathematical physics. We mainly try to present an application of Exp-function method and offer solutions for common errors wich occur during some of the recent works.

**Keywords:**
Exp-function method,
Rosenau Kawahara equation,
Rosenau Korteweg-de Vries equation,
nonlinear partial differential
equation.

##### 2505 Solution of Nonlinear Second-Order Pantograph Equations via Differential Transformation Method

**Authors:**
Nemat Abazari,
Reza Abazari

**Abstract:**

In this work, we successfully extended one-dimensional differential transform method (DTM), by presenting and proving some theorems, to solving nonlinear high-order multi-pantograph equations. This technique provides a sequence of functions which converges to the exact solution of the problem. Some examples are given to demonstrate the validity and applicability of the present method and a comparison is made with existing results.

**Keywords:**
Nonlinear multi-pantograph equation,
delay differential equation,
differential transformation method,
proportional delay conditions,
closed form solution.

##### 2504 An Interval Type-2 Dual Fuzzy Polynomial Equations and Ranking Method of Fuzzy Numbers

**Authors:**
Nurhakimah Ab. Rahman,
Lazim Abdullah

**Abstract:**

According to fuzzy arithmetic, dual fuzzy polynomials cannot be replaced by fuzzy polynomials. Hence, the concept of ranking method is used to find real roots of dual fuzzy polynomial equations. Therefore, in this study we want to propose an interval type-2 dual fuzzy polynomial equation (IT2 DFPE). Then, the concept of ranking method also is used to find real roots of IT2 DFPE (if exists). We transform IT2 DFPE to system of crisp IT2 DFPE. This transformation performed with ranking method of fuzzy numbers based on three parameters namely value, ambiguity and fuzziness. At the end, we illustrate our approach by two numerical examples.

**Keywords:**
Dual fuzzy polynomial equations,
Interval type-2,
Ranking method,
Value.