Wavelet Theory: Applications of the Wavelet

In this article, the approxmated solutions of ordinary differential equations of fractional order using Haar wavelet and B-spline bases are introduced. The algorithm of collection method is updated using two basis. Several initial value problems has been solved to show the applicability and efficacy of the Haar wavelet and B-spline basis. An application of Lane-Eman equation has been introduced and studied. The approximated results have clearly shown the advantage and the efficiency of the modified method in terms of accuracy and computational time.

The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0⩽t⩽2 and 1<α⩽2.

Periodic Solutions in the Golden Sixties: The Birth of a Continuation Theorem

Introduction Suggestions for the readerSuggestions for the reader Fredholm mappings of index zero and linear boundary value problems Degree theory for some classes of mappings Duality theorems for several fixed point operators associated to periodic problems for ordinary differential equations Existence theorems for equations in normed spaces Boundary value problems for second order nonlinear vector differential equations Periodic solutions of ordinary differential equations with one-sided growth restrictions Bound sets for functional differential equations The index of isolated zeros of some mappings Bifurcation theory Periodic solutions of autonomous ordinary differential equations around an equilibrium References.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

In this article, a technique called Haar wavelet-Picard technique is proposed to get the numerical solutions of nonlinear differential equations of fractional order. Picard iteration is used to linearize the nonlinear fractional order differential equations and then Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of Picard iteration, solution is updated by the Haar wavelet method. The results are compared with the exact solution. &nbsp; Key words: Fractional differential equations, Wavelet analysis, Caputo derivative, Haar wavelets, Picard iteration.

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one‐sided estimate type that is formulated in terms of so‐called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1‐complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so‐called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

Recently, wavelet analysis or wavelet transform has been developed as a mathematical instrument for many problems. In the present paper, the numerical technique is based on Haar wavelet collocation points and operational matrices over a new interval [-ϑ,ϑ]. So, we have established a new operational matrix of integration the Haar wavelets functions for a chosen domain from -ϑ to 0. The operational matrices of integration and product are utilized to convert the linear ordinary differential equations together with variable coefficients into algebraic matrix equation which can be resolved by MATLAB. Three numerical examples have been displayed which involving first order differential equations including variable coefficients. The results illustrate that the suggested approach is completely reasonable in comparison with correct solution. Moreover, the exactness of the variable can be enhanced by increasing the Haar wavelets resolution.

Differential equation plays an important role in understanding various phenomenon of science and engineering. Various methods have been developed by mathematicians to obtain a numerical solution of differential equations with minimum errors, transform methods are one of such numerical methods. Till present no transform methods claims to have zero error in obtaining the numerical or exact solution of the given differential equations. Few of well-known transform methods are Laplace transform, Sumudu transforms, Differential transform, Integral transform method etc. In this paper the review of Laplace transform is presented which is one of the tools used by scientists and researchers in finding the solution to their differential equation. In this work, 18 research papers of applications of Laplace transform have been studied and it is shown how this transform has been used by other researchers to obtain the exact solution of ordinary, partial and fractional differential equations. To present a literature review on application of Laplace transform is the main objective of this work.

Dedicated to Professor Yuzan He on the Occasion of his 80th BirthdayIn this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley &amp; Sons, Ltd.

This paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second‐order Sturm–Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.

PurposeThe purpose of this paper is the comparative analysis of Haar Wavelet Method and Optimal Homotopy Asymptotic Method for fractional Fisher type equation. In this paper, two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), have been presented. The Haar wavelet method is an efficient numerical method for the numerical solution of fractional order partial differential equation like the Fisher type. The approximate solutions of the fractional Fisher-type equation are compared with those of OHAM and with the exact solutions. Comparisons between the obtained solutions with the exact solutions exhibit that both the featured methods are effective and efficient in solving nonlinear problems. However, the results indicate that OHAM provides more accurate value than the Haar wavelet method.Design/methodology/approachComparisons between the solutions obtained by the Haar wavelet method and OHAM with the exact solutions exhibit that both featured methods are effective and efficient in solving nonlinear problems.FindingsThe comparative results indicate that OHAM provides a more accurate value than the Haar wavelet method.Originality/valueIn this paper, two reliable techniques, the Haar wavelet method and OHAM, have been proposed for solving nonlinear fractional partial differential equation, i.e. fractional Fisher-type equation. The proposed novel methods are well suited for only nonlinear fractional partial differential equations. It also exhibits that the proposed method is a very efficient and powerful technique in finding the solutions for the nonlinear time fractional differential equations. The main significance of the proposed method is that it requires less amount of computational overhead in comparison to other numerical and analytical approximate methods. The application of the proposed methods for the solutions of time fractional Fisher-type equations satisfactorily justifies its simplicity and efficiency.

The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations

Periodic solutions of delay differential equations (DDEs) splitting into stable and unstable branches are examined in an infinite-dimensional space for fixed and multiple time delays. The center manifold theorem and the classical Hopf bifurcation theorem for the study of periodic solutions of ordinary differential equations (ODEs) are employed to reduce the infinite-dimensional character of the DDEs to finite-dimensional ODEs. Using integral averaging method, the vector field of the ODEs is converted and averaged into amplitude a and phase φ relations. From these relations bifurcation equations of the form ℑ(a, φ) = 0 for the solution branches are derived.

Ordinary differential equations (ODEs) are a fundamental tool for modeling dynamical systems in various scientific fields. However, solving ODEs analytically can be challenging, and numerical methods can be computationally expensive or inaccurate. In recent years, deep neural networks (DNNs) have emerged as a promising alternative for solving ODEs. This paper focuses on the utilization of DNNs to solve first and second-order ODEs. Existing works in this field are reviewed, with an emphasis on identifying their limitations. Subsequently, an approach is proposed to overcome these limitations. The proposed method utilizes a combination of a deep neural network and various activation functions to achieve accurate and efficient solutions for ordinary differential equations (ODEs). Deep neural networks (DNNs) have shown significant potential in solving ordinary differential equations (ODEs) due to their ability to approximate complex functions and their capacity to learn from data. Traditional numerical methods for solving ODEs have some limitations such as requiring large amounts of computation, limited accuracy, and difficulty in handling high-dimensional systems. The aim is to overcome these limitations by leveraging the capabilities of DNNs, that have the potential to learn and generalize from data, provide more accurate solutions, requiring fewer computational resources, and being able to handle high-dimensional systems. This paper aims to explore the use of DNNs in solving 1st and 2nd order differential equations, and proposes a novel approach for improving the accuracy and efficiency of the solution process.