Computational Methods for Differential EquationsComputational Methods for Differential Equations
https://cmde.tabrizu.ac.ir/
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https://cmde.tabrizu.ac.ir/
Feed provided by Computational Methods for Differential Equations. Click to visit.A numerical method for partial fractional Fredholm integro-differential equations
https://cmde.tabrizu.ac.ir/article_8588_1088.html
‎‎‎‎In this paper‎, ‎an expansion method based on orthonormal polynomials is presented to find the numerical solution of partial fractional Fredholm integro-differential equations (PFFIDEs)‎. ‎A PFFIDE is converted to a system of linear algebraic equations‎, ‎which is solved for the expansion coefficients of approximate solution based on orthonormal polynomials‎. ‎An estimation error is discussed and some illustrative examples are given to demonstrate the validity and applicability of the proposed method‎.Sun, 31 Mar 2019 19:30:00 +0100Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of ...
https://cmde.tabrizu.ac.ir/article_8590_1088.html
In this paper, a numerical method for solving a class of nonlinear mixed Fredholm-Volterra integro-diﬀerential equations of fractional order is presented. The method is based upon Bernoulli wavelets approximations. The operational matrix of fractional order integration for Bernoulli wavelets is utilized to reduce the solution of the nonlinear fractional integro-diﬀerential equations to system of algebraic equations. Illustrative examples are included to demonstrate the eﬃciency and accuracy of the method.Sun, 31 Mar 2019 19:30:00 +0100Hermite wavelets method for the numerical solution of linear and nonlinear singular initial and ...
https://cmde.tabrizu.ac.ir/article_8675_1088.html
In this article, Modified Hermite wavelets based numerical method is developed for the solution of singular initial and boundary value problems. It consists of reducing the differential equations with the associated initial and boundary conditions into system of algebraic equations by expanding the unknown function as a series of Hermite wavelets with unknown coefficients. Obtained system of equations are solved using Newton’s iterative method through Matlab. Illustrative examples are considered to demonstrate the applicability and accuracy of the proposed technique. Obtained results are compared favorably with the exact solutions. Also, we proved the theorem reveals that, when exact solution can be obtained by the proposed method.Sun, 31 Mar 2019 19:30:00 +0100Factorization method for fractional Schrödinger equation in D-dimensional fractional space and ...
https://cmde.tabrizu.ac.ir/article_8592_1088.html
In this paper, we consider a $D$-dimensional fractional Schr"odinger equation with a Coulomb potential. By using the associated Laguerre and Jacobi equations, we obtain the wave function and energy spectrum and this then enable us to separate this equation in terms of the radial and angular momentum parts respectively. Also the associated Laguerre and Jacobi equations makes it possible to further factorize the $D$-dimensional fractional Schr"odinger equation such that the resulting equations can be expressed in terms of the first order operators which are basically raising and lowering operators.Sun, 31 Mar 2019 19:30:00 +0100Accurate splitting approach to characterize the solution set of boundary layer problems
https://cmde.tabrizu.ac.ir/article_8666_1088.html
The boundary layer (BL) is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. This paper studies singularly perturbed fractional differential equations where the fractional derivatives are defined in the Caputo sense. The solution of such equations, with appropriate boundary conditions, displays BL behavior. The solution out of the BL is estimated by the solution of the reduced problem and the layer solution is approximated by means of a modified truncated Chebyshev series. The coefficients of the truncated series are evaluated using a novel operational matrix technique. Moreover, the stability and the error analysis of the proposed method are analyzed. Several examples illustrate the validity and applicability of the method.Sun, 31 Mar 2019 19:30:00 +0100On using topological degree theory to investigate a coupled system of non linear hybrid ...
https://cmde.tabrizu.ac.ir/article_8589_1088.html
In this work we discuss the existence of solutions of nonlinear fractional differential equations. By using the topological degree theory, some results on the existence of solutions are obtained. Our analysis relies on the reduction of the problem considered to the equivalent system of Fredholm integral equations. As applications, an examples is also provided to illustrate our main results.Sun, 31 Mar 2019 19:30:00 +0100Interval structure of Runge-Kutta methods for solving optimal control problems with uncertainties
https://cmde.tabrizu.ac.ir/article_8686_1088.html
In this paper, a new interval version of Runge-Kutta methods is proposed for time discretization and solving of optimal control problems (OCPs) in the presence of uncertain parameters. A new technique for interval arithmetic is introduced to evaluate the bounds of interval functions. The proposed approach is based on the new forward representation of Hukuhara interval differencing and combining it with Runge-Kutta method for solving the OCPs with interval uncertainties. To perform the proposed method on OCPs, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, using some algebraic manipulations, they are converted to an ordinary differential equation to achieve the interval optimal solution for the considered OCP with uncertain parameters. Shooting method is also employed to cover the Runge-Kutta methods restrictions in solving the OCPs with boundary values. The simulation results are applied to some practical case studies to demonstrate the effectiveness of the proposed method.Sun, 31 Mar 2019 19:30:00 +0100Linear fractional fuzzy differential equations with Caputo derivative
https://cmde.tabrizu.ac.ir/article_8685_1088.html
In this paper, we study linear fractional fuzzy differential equations involving the Caputo generalized Hukuhara derivative. Using the fuzzy Laplace transform, we present the general form of solutions in terms of Mittag-Leffler functions. Finally, some examples are provided to illustrate our results.Sun, 31 Mar 2019 19:30:00 +0100Approximate symmetry and exact solutions of the perturbed nonlinear Klein-Gordon equation
https://cmde.tabrizu.ac.ir/article_8562_1088.html
In this paper, the Lie approximate symmetry analysis is applied to investigate new exact solutions of the perturbed nonlinear Klein-Gordon equation. The nonlinear Klein-Gordon equation is used to model many nonlinear phenomena. The tanh-coth method, is employed to solve some of the obtained reduced ordinary differential equations. As a result, we construct new analytical solutions with small parameter which is effectively obtained by the proposed method.Sun, 31 Mar 2019 19:30:00 +0100Analysis of meshless local radial point interpolant on a model in population dynamics
https://cmde.tabrizu.ac.ir/article_8591_1088.html
In this work, we present an improvement of the spectral meshless radial point interpolation (SMRPI) method to uncover a simulation behaviour of the population dynamic model which mathematically is the nonlinear partial integro-differential equation. This PDE is a kind of competition strategy in which equivalent individuals match for the same supplies. oreover, this boundary value problem is a particular type of reaction-diffusion problem augmented to an integral term corresponding to the nonlocal consumption of resources. As a result of applying meshless method, it does not matter how the geometry of the domain is complicated because the method enjoys the element free adoption. Applying the SMRPI on the two-dimensional integral equation leads to a linear system of algebraic equations which is easy to treat. Finally, some numeric experiments are presented to show the reliable results.Sun, 31 Mar 2019 19:30:00 +0100Spectral solution of fractional fourth order partial integro-differential equations
https://cmde.tabrizu.ac.ir/article_8691_1088.html
In this paper, mixed spectral method is applied to solve the fractional fourth order partial integro-differential equations together with weak singularity. Eigenfunctions of the fourth order self-adjoint positive-definite differential operator are used for the discretization of spatial variable and its derivatives. Also, shifted Legendre polynomials are applied to the discretization of time variable. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. The method is easy to apply and produces very accurate numerical results.Sun, 31 Mar 2019 19:30:00 +0100Stabilization of linear systems of delay differential equations by the delayed feedback method
https://cmde.tabrizu.ac.ir/article_8553_1088.html
This paper consists of two folds. At first, we deal with the stability analysis of a linear system of delay differential equations. It is shown that the direct and cluster treatment methods are not applicable if there are some purely imaginary roots of the characteristic equation with multiplicity greater than one. To overcome the above difficulty, the system is decomposed into several subsystems. For the decomposition of a system, an invertible transformation is required to convert the matrices of the system into a block triangular (diagonal) form simultaneously. To achieve this goal, a necessary and sufficient condition is established. The second part concerns the stabilization of a linear system of delay differential equations using the delayed feedback method and design a controller for generating the desired response. More precisely, the unstable poles of the linear system of delay differential equations are moved to the left-half of the complex plane by the delayed feedback method. It is shown that the performance of the linear system of delay differential equations can be improved by applying the delayed feedback method.Sun, 31 Mar 2019 19:30:00 +0100