In this study, we propose a novel Haar wavelet-based Galerkin method to solve nonlinear optimal control problems with applications to unmanned vehicle navigation. The method addresses the critical challenge of optimizing energy consumption while ensuring safe navigation in dynamic environments with multiple moving obstacles. By leveraging the computational efficiency and scalability of Haar wavelets, combined with the robustness of the Galerkin approach, we demonstrate convergence to the optimal solution under feasibility and consistency conditions. Comprehensive numerical simulations, including diverse and complex obstacle scenarios, validate the method's practicality. Through detailed trajectory, speed, and direction analyses, we highlight the approach's ability to adapt to real-world navigation challenges, making it a promising tool for autonomous system optimization.

First, this paper defines a general nonlinear optimal control problem with state/control constraints and its approximation problem as the Haar wavelet Galerkin optimal control problem (HWGOCP). Then, a Haar wavelet-based Galerkin numerical method has been developed, which converts it to a nonlinear optimization problem. We theoretically prove that a Haar wavelet feasible solution of HWGOCP will exist. We also show that the approximate solutions of HWGOCP are consistent and converge to the optimal solution of the problem. A variety of application problems have been considered, which include optimal control of tumour growth using Chemotherapy drugs, optimal control of infection via the SIS model using treatment, the Brachistochrone problem in mechanics, optimal control of mold using a fungicide, optimal control of pH value of a chemical reaction to determine the quality of a product, etc.

Strongly singular integral equations of order four have applications in fracture mechanics, and Gegenbauer polynomials have never been used to solve these equations. This motivated us to develop a Gegenbauer polynomial-based Galerkin method to solve a singular integral equation of order four. We first prove the problem's well-posedness. Then, we show the theoretical convergence of the numerical scheme and derive the rate of convergence and the error estimates. We validate the theoretical error estimates numerically in test examples. We implement the proposed method to a crack problem and compare it with existing results in the literature.

In this study, we present a novel indirect Galerkin method based on Haar wavelets for solving nonlinear optimal control problems, a domain characterized by its complex and challenging dynamics. By leveraging Pontryagin’s Principle, we derive the first-order necessary conditions and reformulate the problem as a two-point boundary value problem. The proposed method integrates Haar wavelets with the Galerkin method to approximate the state, control, and costate variables, providing a computationally efficient framework. The method benefits from the inherent properties of Haar wavelets, such as their ability to handle sharp discontinuities and abrupt changes, which make them highly suitable for this application.To evaluate the performance of the proposed approach, we validate it against a series of benchmark problems. The numerical results demonstrate superior accuracy and convergence when compared to existing methods, such as the Haar wavelet collocation and pseudospectral approaches. Furthermore, the cost functional values achieved by the method consistently outperform those of previously reported techniques, highlighting its practical efficacy. Despite the computational challenges posed by solving larger systems of equations, the method remains robust and effective across various problem settings. Future research directions include enhancing the theoretical framework by establishing rigorous convergence proofs and error bounds. Overall, this work significantly advances the numerical methodologies available for nonlinear optimal control problems, contributing a powerful tool for addressing real-world applications.

Millions of people are infected by Dengue every year. So to understand its mechanism, we have considered one of the recent fractional and nonlinear Dengue models proposed by Diethelm. However, the problem is that its exact solution is not known. Therefore, an extended Haar wavelets based numerical method has been implemented to solve it. Although Haar wavelets-based numerical methods have been used by researchers in recent years its theoretical error analysis over the extended interval [0, a), a is an element of R+, has never been done before for fractional dynamical systems. It helps to show the convergence of a sequence of approximate solutions to the exact solution. Also, the derived error bound gives a guarantee that error in the considered Haar approximate solution cannot exceed its bound. And any desired accuracy can be obtained by increasing the number of Haar wavelets in the approximate solution.

In this paper, we develop Galerkin’s residual-based numerical scheme for solving a system of Cauchy-type singular integral equations of index minus N using Chebyshev polynomials of the first and second kind, where N is the total number of Cauchy-type singular integral equations in the system. Without theoretical analysis, a numerical scheme is not justified. Therefore, first, we prove the well-posedness of the system of Cauchy-type singular integral equations with the help of the compactness of an operator. Further, we derive a theoretical error bound and the order of convergence. Also, we show that the resulting system of equations obtained by applying the algorithm is well-posed together with an explicit representation of the solution in matrix form. Finally, we give some illustrative examples to validate the theoretical error bounds numerically.

A system of hypersingular integral equations gets formed while solving curved crack or multi-crack problems in fracture mechanics. However, the issue is that its analytical solution is only available for some particular cases. For this, we develop a residual-based Galerkin’s numerical method using Chebyshev polynomials of the second kind with its complete analysis. We prove the system’s well-posedness and the theoretical and numerical convergence of the approximation scheme. We derive the convergence rate, validate error bounds via test examples, and compare them with the Legendre polynomial-based numerical method in the literature.

Haar wavelet-based numerical algorithms have recently been developed for various mathematical problems, including optimal control problems. However, no numerical algorithm is complete without its theoretical analysis. In this paper, we have shown the consistency and feasibility of the Haar wavelet-based collocation method for solving nonlinear optimal control problems that have a single state and a single control variable with constraints. The accuracy of the method has been shown through some application problems.

This paper propose a residual based Galerkin method with Legendre polynomial as a basis functions to find the approximate solution of hypersingular integral equations. These equations occur quite naturally in the field of aeronautics such as problem of aerodynamics of flight vehicles and during mathematical modeling of vortex wakes behind aircraft. The analytic solution of these kind of equations is known only for a particular case ( m(x,t) =0 in Eqn (1)). Also, in these singular integral equations which occur during the formulation of many boundary value problems, the known function m(x,t) in (Eqn (1)) is not always zero. Our proposed method find the approximate solution by converting the integral equations into a linear system of algebraic equations which is easy to solve. The convergence of sequence of approximate solutions is proved and error bound is obtained theoretically. The validation of derived theoretical results and implementation of method is also shown with the aid of numerical illustrations.

Hepatitis C is a liver disease which is caused by the hepatitis C virus. The virus can cause both acute and chronic hepatitis. As per the fact sheet of WHO updated on October 2017 for hepatitis C, 71 million people have been estimated for chronic hepatitis C infection in the whole world. Approximately 399,000 people die each year from hepatitis C. No vaccine for hepatitis C is available currently and research is going on in all possible directions. In order to control the hepatitis C virus and to better understand its mechanism, researchers have been proposing nonlinear integer as well as fractional mathematical models. But in general, no analytical method is available to solve these. In this paper, a Haar wavelet based numerical method has been implemented to find an approximate solution of a nonlinear fractional model for hepatitis C proposed by Ahmed and El-Saka [1, 2]. The error bounds have also been calculated.

In this paper, the problem of finding numerical solution for a system of Cauchy type singular integral equations of first kind with index zero is considered. The analytic solution of such system is known. But it is of limited use as it is a nontrivial task to use it practically due to the presence of singularity in the known solution itself. Therefore, a residual based Galerkin method is proposed with Legendre polynomials as basis functions to find its numerical solution. The proposed method converts the system of Cauchy type singular integral equations into a system of linear algebraic equations which can be solved easily. Further, Hadamard conditions of well-posedness are established for system of Cauchy singular integral equations as well as for system of linear algebraic equations which is obtained as a result of approximation of system of singular integral equations with Cauchy kernel. The theoretical error bound is derived which can be used to obtain any desired accuracy in the approximate solution of system of Cauchy singular integral equations. The derived theoretical error bound is also validated with the help of numerical examples.

A system of hypersingular integral equations occurs quite naturally in several branches of science and engineering during the formulation of many boundary value problems. The analytical solution is known for the system of dominant equations. However, there are many real world problems such as crack problems occur in the field of fracture mechanics which may not be formulated as system of dominant equations. Therefore, we propose a numerical method to find the approximate solution for such generalized form. The convergence of the proposed method is proved in LN2 space. This convergence helps to derive theoretical error bound for the error between the exact and the approximate solution. An application of the proposed method in finding numerical solution of hypersingular integral equation over the curves is shown. Finally, the derived theoretical error bound is numerically calculated and validated with the help of numerical examples.

In this paper, a Haar wavelets based numerical method to solve a system of linear or nonlinear fractional differential equations has been proposed. Numerous nontrivial test examples along with practical problems from fluid dynamics and chemical engineering have been considered to illustrate applicability of the proposed method. We have derived a theoretical error bound which plays a crucial role whenever the exact solution of the system is not known and also it guarantees the convergence of approximate solution to exact solution.

Throughout the world people have been suffering from the infectious diseases like Rubella, Herpes Simplex, Hepatitis B, Chagas, and HIV(AIDS) which have been causing loss of millions of lives and billions of rupees in cure. These diseases get transmitted via both horizontal and vertical transmission routes. One among the most recent mathematical models for modeling the above infectious diseases is fractional order nonlinear SEIR model 131 with non-constant population. In this paper, we have proposed a computationally faster and simpler numerical method based on Haar wavelets to solve the SEIR model. The error bounds have also been derived and validated. (C) 2017 Elsevier B.V. All rights reserved.

In this paper, a numerical method is proposed to solve the Fredholm-Volterra fractional integro-differential equation with nonlocal boundary conditions by using Haar wavelets. A collocation based Galerkin's method is applied by using Haar wavelets as basis functions over the interval [0, 1). It converts the Fredholm-Volterra fractional integro-differential equation into a system of m linear equations. On incorporating q nonlocal boundary conditions, it leads to further q equations. All together it will give a system of (m + q) linear equations in (m + q) variables which can be solved. A variety of test examples are considered to illustrate the proposed method. The actual error is also measured with respect to a norm and the results are validated through error bounds.

Cauchy type singular integral equations with index zero naturally occur in the field of aerodynamics. Literature is very much developed for these equations and Chebyshevs polynomials are most frequently used to solve these integral equations. In this paper, a residual based Galerkins method has been proposed by using Legendre polynomial as basis functions to solve Cauchy singular integral equation of index zero. It converts the Cauchy singular integral equation into system of equations which can be easily solved. The test examples are given for illustration of proposed numerical method. Error bounds are derived as well as implemented in all the test examples.

In this paper, a numerical method has been proposed to find an approximate solution of Cauchy type singular integral equations of first kind. Legendre polynomials have been used as basis functions. The effectiveness of the method is shown with the help of various test examples. Moreover, we have shown the application of our proposed method to solve a fracture mechanics problem which occurs in an infinite isotropic elastic medium with constant load σ along its four branches. The obtained results are in good agreement with those already present in literature. AMS (MOS) Subject Classification. 41A10, 45A05, 45E05.

In this paper, a robust but very simple numerical method is developed to solve various cases of Cauchy type singular integral equation. For this, first Bernstein polynomials are defined which are used for approximation of solution of the given singular integral equation. Then numerical method is introduced by using Bernstein polynomials. This ultimately leads to solution of a system of linear algebraic equations. Examples are illustrated to demonstrate simplicity of proposed method. Results are also compared with those present in literature to claim better efficiency of the method introduced. (C) 2013 Elsevier Inc. All rights reserved.

In this paper a numerical method is developed to solve the Bagley-Torvik equation. The Bagley-Torvik equation is a fractional differential equation which occurs quite frequently in various branches of Applied Mathematics and Mechanics. The solution to this equation is proposed by using second kind Chebyshev wavelet. It finally reduces the equation in to the system of linear equations which can be easily solved. Examples are illustrated to demonstrate simplicity of proposed method. Results are also compared with those present in literature.