Solution of the Systems of Delay Integral Equations in Heterogeneous Data Communication through Haar Wavelet Collocation Approach

In this work, the Haar collocation scheme is used for the solution of the class of system of delay integral equations for heterogeneous data communication. The Haar functions are considered for the approximation of unknown function. By substituting collocation points and applying the Haar collocation technique to system of delay integral equations, we have obtained a linear system of equations. For the solution of this system, an algorithm is developed in MATLAB software. The method of Gauss elimination is utilized for the solution of this system. Finally, by using these coefficients, the solution at collocation points is obtained. The convergence of Haar technique is checked on some test problems.

A powerful numerical technique for treating twelfth-order boundary value problems

In this article, a fast algorithm is developed for the numerical solution of twelfth-order boundary value problems (BVPs). The Haar technique is applied to both linear and nonlinear BVPs. In Haar technique, the twelfth-order derivative in BVP is approximated using Haar functions, and the process of integration is used to obtain the expression of lower-order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking the convergence of the proposed technique. A comparison of the results obtained by the present technique with results obtained by other techniques reveals that the present method is more effective and efficient. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The convergence rate using different numbers of collocation points is also calculated, which is approximately equal to 2.

Efficient Numerical Scheme for the Solution of Tenth Order Boundary Value Problems by the Haar Wavelet Method

In this paper, an accurate and fast algorithm is developed for the solution of tenth order boundary value problems. The Haar wavelet collocation method is applied to both linear and nonlinear boundary value problems. In this technqiue, the tenth order derivative in boundary value problem is approximated using Haar functions and the process of integration is used to obtain the expression of lower order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking validation and the convergence of the proposed technique. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The experimental rate of convergence using different number of collocation points is also calculated, which is nearly equal to 2.

The Generalized Haar Spaces and Their Adaptive Decomposition

This paper is devoted to the numericalinformation flows and adaptive decompositions of thegeneral Haar functions connected with them. The aim ofthis paper is to propose an adaptive wavelet decompositionusing an adaptive compression algorithm for a flow ofnumerical information of length M with complexity𝑶(𝑴) and with a given precision of 𝜀> 0. The numericalflows are associated with irregular spline grids. This paperdiscusses the calibration relations, the embedding of thegeneral Haar spaces and their wavelet decompositions.The structure of the decomposition/ reconstructionalgorithms are done. The cases of the finite and the infiniteflows are considered. The paper discusses various methodsof adaptive Haar approximations for the flow of functionvalues. Assuming that the values of the first derivative ofthe approximated function is known (exactly orapproximately), the complexity of using an adaptive grid isestimated for a priori specified approximation accuracy.The number of K knots in the adaptive grid determine therequired amount of memory for storage of thecompression results. The number of M knots of the initialgrid characterizes the number of operations required toobtain the adaptive compression. In the case of access tothe derivative values (or their approximations) thenumber of digital operations is proportional to the numberM. If it does not have access to the last ones then thenumber of required operations has the order of M2 (in thegeneral case). If additionally, the approximated flow isconvex, then the number of required operations has theorder of M log2M. In all cases the result requires thecomputer memory amount to be of the order of K

The special atom space and Haar wavelets in higher dimensions

In this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spaces such as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higher dimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.

Solving Two-Dimensional Nonlinear Volterra Integral Equations Using Rationalized Haar Functions in the Complex Plane

In this work, two-dimensional rational Haar wavelet method has been used to solve the twodimensional Volterra integral equations. By using fixed point Banach theorem we achieved the order of convergence and the rate of convergence is O(n(2q)n). Numerical solutions of three examples are presented by applying a simple and efficient computational algorithm.