Integro-differential Equations Of Second Kind Research Articles

Numerical Investigation with Convergence and Stability Analyses of Integro-Differential Equations of Second Kind

In this paper, integro-differential equations are solved by using an efficient numerical technique, namely, Multistage Optimal Homotopy Asymptotic method. The existence and uniqueness of solutions are established by the application of Lipschitz condition. Convergence of approximate solutions along with stability are also carried out. Some examples are solved to highlight the vital characteristics of the applied numerical scheme. Error estimation and comparison of derived results with existing exact solutions and those results which already available in the literature through graphical illustrations and tables reveal that Multistage Optimal Homotopy Asymptotic algorithm is more efficient and fruitful.

Projection methods for approximate solution of a class of nonlinear Fredholm integro-differential equations

The aim of this article is to find the approximate solution of the nonlinear Fredholm integro-differential equations of second kind with smooth kernels with less computational complexity and investigate the asymptotic behavior of convergence of the approximate solutions by using global polynomials based projection methods. We develop the theoretical framework for the nonlinear Fredholm integro-differential equations to obtain the superconvergence results by Legendre polynomial based projection methods and their iterated versions. Numerical examples are considered to demonstrate the theoretical results.

On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis

In this work we consider a class of fractional order Volterra integro-differential equations of first kind where the fractional derivative is considered in the Caputo sense. Here, we consider the initial value problem and the boundary value problem separately. For simplicity of the analysis, we reduce each of these problems to the fractional order Volterra integro-differential equation of second kind by using the Leibniz’s rule. We have obtained sufficient conditions for the existence and uniqueness of the solutions of initial and the boundary value problems. An operator based method has been considered to approximate their solutions. In addition, we provide a convergence analysis of the adopted approach. Several numerical experiments are presented to support the theoretical results.

Fifth Order Multistep Block Method for Solving Volterra Integro-Differential Equations of Second Kind

In the present paper, the multistep block method is proposed to solve the linear and non-linear Volterra integro-differential equations (VIDEs) of the second kind using constant step size. The proposed block method of order five consists of two point block method presented as in the simple form of Adams Moulton type. The numerical solutions are obtained at two new values simultaneously at each of the integration step. In VIDEs, the unknown function appears in the form of derivative and under the integral sign. The approximation of the integral part is estimated using the Boole’s quadrature rule. The stability region is shown, and the numerical results are presented to illustrate the performance of the proposed method in terms of accuracy, total function calls and execution times compared to the existing method.

Fuzzy Volterra Integro-Differential Equations Using General Linear Method

In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The derivation of general linear method is based on the theory of B-series and rooted trees. Here, the fuzzy general linear method using the approach of generalized Hukuhara differentiability and combination of composite Simpson’s rules together with Lagrange interpolation polynomial is constructed for numerical solution of fuzzy volterra integro-differential equations. To illustrate the performance of the method, the numerical results are compared with some existing numerical methods.

CONVERGENCE OF THE SBA ALGORITHM APPLYING TO SOLVE VOLTERRA NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF SECOND KIND

In this paper we shwo the convergence of the SBA method on the Volterranonlinear integro-differential equations of second kind.

Numerical Solutions of Non-linear Fredholm Integro-Differential Equations of Second Kind

In this paper, we propose a method that determines the approximate solution of nonlinear Fredholm integro-differential equations of second kind. The integro-differential equation is converted into an integral equation by using suitable transformation. The integral part is approximate by Newton Cote's method. The convergence of method is established. The theory is supported by some examples.

A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.

Numerical solution for system of singular nonlinear Volterra integro-differential equations by Newton-Product method

In this paper, we apply Newton-Product method to obtain the numerical solutions of singular nonlinear Volterra integro-differential equations of second kind. We applied Newton’s method to linearize it and then we found product integration solution of the linear form. Sufficient conditions for convergence of the numerical method are given and their applicability is illustrated with some examples.

Modified Sumudu-Adomian method applied to nth-order Volterra integro-differential equations

In this work, we introduced a modified approach of Sumudu-Adomian method. In this approach, we combined the Sumudu transform method with the modified Adomian method. To study the performance of this approach, we applied the modified technique for solving selected Volterra integro-differential equations of second kind and compared our results with the results obtained by the Sumudu-Adomian method. This study showed the capability, simplicity and effectiveness of the modified approach. It also showed that the modified approach reduces the computational difficulties compared with the Sumudu-Adomian method.

Conditions for bounded solutions of non-Markovian quantum master equations

For the most general quantum master equations, also called Nakajima–Zwanzig or non-Markovian equations, we define suitable boundedness conditions on integral kernels and inhomogeneity terms in order to derive with mathematical rigor an upper bound on solutions, as required by the von Neumann conditions. Such equations are of importance for quantum dynamics of open systems with arbitrary couplings to environment and arbitrary entangled initial states. The derivation is based on an equivalent coherence-vector representation in finite dimension n leading to coupled Volterra integro-differential equations of second kind and convolution type in an (n 2−1)-dimensional real vector space. As examples, analytical and numerical model solutions are worked out for 2-level systems in order to test suitable trial functions for input quantities. All this is motivated by the fact that exact solutions can hardly be found but appropriate trial functions may provide a reasonable semiphenomenological description of complicated quantum dynamics.

On linear Fredholm integro-differential equations of first kind

We study the question whether linear one-dimensional integro-differential equations with constant limits of integration (equations of Fredholm type) containing no free differential expression (equations of first kind) can be reduced to integral equations of first kind and to Fredholm integro-differential equations of second kind.

Plane viscous incompressible flow in regions with a free boundary

We consider the deformation of a fluid body under the action of surface tension. The apparatus of hydrodynamic potentials is applied to reduce the problem to integrodifferential equations of second kind. An algorithm is constructed that determines the deformation of the fluid body successively in time. Results of numerical calculations are reported. In particular, the problem of deformation of a fluid ellipse under the action of surface tension is analyzed.