Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.     

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill-posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill-posed problem into well-conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill-posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.     

A numerical assessment of parabolic partial differential equations using Haar and Legendre wavelets

In this paper we present two new numerically stable methods based on Haar and Legendre wavelets for one- and two-dimensional parabolic partial differential equations (PPDEs). This work is the extension of the earlier work ,  and  from one- and two-dimensional boundary-value problems to one- and two- dimensional PPDEs. Two generic numerical algorithms are derived in two phases. In the first stage a numerical algorithm is derived by using Haar wavelets and then in the second stage Haar wavelets are replaced by Legendre wavelets in quest for better accuracy. In the proposed methods the time derivative is approximated by first order forward difference operator and space derivatives are approximated using Haar (Legendre) wavelets. Improved accuracy is obtained in the form of wavelets decomposition. The solution in this process is first obtained on a coarse grid and then refined towards higher accuracy in the high resolution space. Accuracy wise performance of the Legendre wavelets collocation method (LWCM) is better than the Haar wavelets collocation method (HWCM) for problems having smooth initial data or having no shock phenomena in the solution space. If sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, LWCM loses its accuracy in such cases, whereas HWCM produces a stable solution in such cases as well. Contrary to the existing methods, the accuracy of both HWCM and LWCM do not degrade in case of Neumann’s boundary conditions. A distinctive feature of the proposed methods is its simple applicability for a variety of boundary conditions. Performances of both HWCM and LWCM are compared with the most recent methods reported in the literature. Numerical tests affirm better accuracy of the proposed methods for a range of benchmark problems.     

The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis

The two-grid method is a technique to solve the linear system of algebraic equations for reducing the computational cost. In this study, the two-grid procedure has been combined with the EFG method for solving nonlinear partial differential equations. The two-grid FEM has been introduced in various forms. The well-known two-grid FEM is a three-step method that has been proposed by Bajpai and Nataraj (Comput. Math. Appl. 2014;68:2277–2291) that the new proposed scheme is an ecient procedure for solving important nonlinear partial differential equations such as Navier–Stokes equation. By applying shape functions of IMLS approximation in the EFG method, a new technique that is called interpolating EFG (IEFG) can be obtained. In the current investigation, we combine the two-grid algorithm with the IEFG method for solving the nonlinear Rosenau-regularized long-wave (RRLW) equation. In other hand, we demonstrate that solutions of steps 1, 2, and 3 exist and are unique and also we achieve an error estimate for them. Moreover, three test problems in one- and two-dimensional cases are given which support accuracy and efficiency of the proposed scheme.     

Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions

Two-dimensional rationalized Haar (RH) functions are applied to the numerical solution of nonlinear second kind two-dimensional integral equations. Using bivariate collocation method and Newton–Cotes nodes, the numerical solution of these equations is reduced to solving a nonlinear system of algebraic equations. Also, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.     

Analysis of ADER and ADER-WAF schemes

^{We study stability properties and truncation errors of the finite-volumeADER schemes on structured meshes as applied to the linear advectionequation with constant coefficients in one-, two- and three-spatialdimensions. Stability of linear ADER schemes is analysed bymeans of the von Neumann method. For nonlinear schemes, we deducethe stability region from numerical experiments. The truncationerror analysis is carried out for linear ADER schemes in one-,two- and three-space dimensions and for nonlinear ADER schemesin one-space dimension.}     

A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique

In this paper, a computational scheme is proposed to estimate the solution of one- and two-dimensional Fredholm-Hammerstein integral equations of the second kind. The method approximates the solution using the discrete Galerkin method based on the moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin technique for integral equations results from the numerical integration of all integrals in the system corresponding to the Galerkin method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The implication of the scheme for solving two-dimensional integral equations is independent of the geometry of the domain. The new method is simple, efficient and more flexible for most classes of nonlinear integral equations. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.     

An effective treatment of nonlinear differential equations with linear boundary conditions using the homotopy analysis method

In this paper, we provide an effective technique to treat nonlinear differential equations with linear boundary conditions that are reduced from a nonlinear problem describing the steady-state boundary-layer flow of a micropolar fluid near the forward stagnation point of a two-dimensional plane surface. The analytical approximations with high accuracy are obtained using the homotopy analysis method, which agree well with the numerical results. This indicates the validity and great potential of the proposed method for solving nonlinear differential equations with linear boundary conditions.     

New computational method for solving some 2-dimensional nonlinear Volterra integro-differential equations

The aim of this paper is to present an efficient numerical procedure for solving the two-dimensional nonlinear Volterra integro-differential equations (2-DNVIDE) by two-dimensional differential transform method (2-DDTM). The technique that we used is the differential transform method, which is based on Taylor series expansion. Using the differential transform, 2-DNVIDE can be transformed to algebraic equations, and the resulting algebraic equations are called iterative equations. New theorems for the transformation of integrals and partial differential equations are introduced and proved. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.     

Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets

In this work, we present a computational method for solving nonlinear Fredholm integral equations of the second kind which is based on the use of Haar wavelets. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples.     

Spectral corrections for Sturm–Liouville problems

The numerical solution of the Sturm–Liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction. In this paper we use the shooting method both for eigenvalues and eigenfunctions. In integrating the corresponding initial value problems we resort to the boundary value method. The technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems. A technique to estimate the eigenvalue errors is also suggested, and seems to be particularly effective for the higher-index eigenvalues. Numerical experiments on some classical Sturm–Liouville problems are presented.     

An Accurate Space–Time Pseudospectral Method for Solving Nonlinear Multi-Dimensional Heat Transfer Problems

In this paper, we consider the numerical solution of the nonlinear one- and two-dimensional heat transfer problems subject to the given initial conditions and linear Robin boundary conditions. We propose a pseudospectral scheme in both time and spatial discretizations for these problems. The discretization processes are constructed through the multi-variate interpolation of the desired solutions in terms of Chebyshev Gauss Lobbato collocation points. Operational matrices of differentiation are constructed via the tensor products for speeding up of the proposed numerical algorithms’ implementation. Some test problems are provided and the numerical simulations are illustrated to show the spectral accuracy in both space and time of the suggested scheme.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method

The main purpose of this article is to describe a numerical scheme for solving two-dimensional linear Fredholm integral equations of the second kind on a non-rectangular domain. The method approximates the solution by the discrete collocation method based on radial basis functions (RBFs) constructed on a set of disordered data. The proposed method does not require any background mesh or cell structures, so it is meshless and consequently independent of the geometry of domain. This approach reduces the solution of the two-dimensional integral equation to the solution of a linear system of algebraic equations. The error analysis of the method is provided. The proposed scheme is also extended to linear mixed Volterra–Fredholm integral equations. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the new technique.     

On <Emphasis Type="Italic">N</Emphasis>-termed approximations in <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript>-norms with respect to the Haar system

In [9], we proved numerically that spaces generated by linear combinations of some two-dimensional Haar functions exhibit unexpectedly nice orders of approximation for solutions of the single-layer potential equation in a rectangle. This phenomenon is closely related, on the one hand, to the properties of the approximation method of hyperbolic crosses and on the other to the existence of a strong singularity for solutions of such boundary integral equations. In the present paper, we establish several results on the approximation for the hyperbolic crosses and on the best N-term approximations by linear combinations of Haar functions in the H ^s -norms, −1 < s < 1/2; this provides a theoretical base for our numerical research. To the author's best knowledge, the negative smoothness case s < 0 was not studied earlier. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

Nonlinear PDE Model for Image Restoration Using Second-Order Hyperbolic Equations

In this article we consider a novel nonlinear PDE-based image denoising technique. The proposed restoration model uses second-order hyperbolic diffusion equations. It represents an improved nonlinear version of a linear hyperbolic PDE model developed recently by the author, providing more effective noise removal results while preserving the edges and other image features. A rigorous mathematical investigation is performed on this new differential model and its well-posedness is treated. Next, a consistent finite-difference numerical approximation scheme is proposed for this nonlinear diffusion-based approach. Our successful image denoising experiments and method comparisons are also described.     

Efficient hybrid method for solving special type of nonlinear partial differential equations

In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two‐ and three‐dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.     

Numerical solutions of partial differential equations by discrete homotopy analysis method

This paper introduces a discrete homotopy analysis method (DHAM) to obtain approximate solutions of linear or nonlinear partial differential equations (PDEs). The DHAM can take the many advantages of the continuous homotopy analysis method. The proposed DHAM also contains the auxiliary parameter ?, which provides a simple way to adjust and control the convergence region of solution series. The convergence of the DHAM is proved under some reasonable hypotheses, which provide the theoretical basis of the DHAM for solving nonlinear problems. Several examples, including a simple diffusion equation and two-dimensional Burgers’ equations, are given to investigate the features of the DHAM. The numerical results obtained by this method have been compared with the exact solutions. It is shown that they are in good agreement with each other.     

Wavelet operational matrix method for solving the Riccati differential equation

A Haar wavelet operational matrix method (HWOMM) was derived to solve the Riccati differential equations. As a result, the computation of the nonlinear term was simplified by using the Block pulse function to expand the Haar wavelet one. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. The capability and the simplicity of the proposed method was demonstrated by some examples and comparison with other methods.     

An orthogonal subspace minimization method for finding multiple solutions to the defocusing nonlinear Schrödinger equation with symmetry

An orthogonal subspace minimization method is developed for finding multiple (eigen) solutions to the defocusing nonlinear Schrödinger equation with symmetry. As such solutions are unstable, gradient search algorithms are very sensitive to numerical errors, will easily break symmetry, and will lead to unwanted solutions. Instead of enforcing a symmetry by the Haar projection, the authors use the knowledge of previously found solutions to build a support for the minimization search. With this support, numerical errors can be partitioned into two components, sensitive versus insensitive to the negative gradient search. Only the sensitive part is removed by an orthogonal projection. Analysis and numerical examples are presented to illustrate the method. Numerical solutions with some interesting phenomena are captured and visualized by their solution profile and contour plots. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013