Hat Functions Research Articles

Solving nonlinear stochastic differential equations via fourth-degree hat functions

The article focuses on introducing novel fourth-degree hat functions (FDHFs) for constructing new operations matrices used to find numerical solutions for nonlinear stochastic differential equations (NSDEs). The technique’s effort is summarized by utilizing FDHFs to construct operations matrices that transform the given problem into nonlinear algebraic equations. The advantage of this technique lies in its simplicity for calculating the unknown coefficients of the function’s approximation without requiring any integration. As a result, the proposed approach incurs low computational expenses. Additionally, an error analysis of this approach was conducted, demonstrating a convergence rate of O(h5). Several examples were implemented to support and illustrate the effectiveness and capability of the proposed technique.

Two new numerical approaches for the fractional distribution of the model of a system of lakes via modified hat and quasi-hat functions

Two new numerical approaches for the fractional distribution of the model of a system of lakes via modified hat and quasi-hat functions

Integro-differential equations: Numerical solution by a new operational matrix based on fourth-order hat functions

This article aims to present a new approach based on the operational matrix method for solving integro-differential equations using numerical techniques. In this method, we employ fourth-degree hat functions (FDHFs) to construct operational matrices. The approach involves two main steps. First, we utilize FDHFs to create operational matrices, which allows us to transform the given problem into a system of algebraic equations. The second step involves solving these algebraic equations numerically. Additionally, we provide an analysis of the errors involved and compare the proposed method with existing techniques. The results demonstrate that the proposed method outperforms its counterparts, highlighting its superiority.

A novel numerical method for solving optimal control problems using fourth-degree hat functions

This paper focuses on solving a class of nonlinear optimal control problems by constructing novel hat functions based on fourth-order polynomials, namely, fourth-degree hat functions (FDHFs). The FDHFs are used in this method to approximate the state equations and cost function. In fact, the FDHFs enable us to turn the optimal control problem under consideration into a nonlinear optimal control problem with unknown coefficients that is easy to solve by any numerical method. The main advantages of the proposed method are its simplicity, ease of application, low computational expense, and avoidance of numerical integration. Several examples have been discussed to demonstrate the efficacy and applicability of the suggested method.

Numerical Solution of Fractional Integro-Differential Equations Via Fourth-Degree Hat Functions

The goal of this paper is to construct new fourth-degree hat functions (FDHFs) and study their properties in order to develop a new numerical method for solving fractional integro-differential equations. The equation under consideration is transformed into a set of algebraic equations by using FDHFs, which makes it simple to solve the system using one of the iterative methods. In fact, this method’s advantage was that it was easy to use and had fifth-order convergence, as we showed in the section on error analysis. The numerical results demonstrate that the new technique works well through the presented examples.

Solving Volterra integral equations via fourth-degree hat functions

The goal of this paper is to develop a novel operation matrix approach to solving Volterra integral equations by constructing fourth-degree hat functions and investigating their properties. This approach requires turning the problem under dissection into a set of algebraic equations and then solving it by any numerical method. In addition, we demonstrate that the convergence of this approach is of the order of O(h5), and numerical results show that it is practical and useful for dealing with such problems.

A numerical algorithm for a class of nonlinear fractional Volterra integral equations via modified hat functions

A numerical algorithm via a modified hat functions (MHFs) has been proposed to solve a class of nonlinear fractional Volterra integral equations of the second kind. A fractional-order operational matrix of integration is introduced. In a new methodology, the operational matrices of MHFs and the powers of weakly singular kernels of integral equations are used as a structure for transforming the main problem into a number of systems consisting of two equations for two unknowns. Relative errors for the approximated solutions are investigated. Convergence analysis of the proposed method is evaluated and convergence rate is addressed. Finally, the extraordinary accuracy of the utilized approach is illustrated by a few examples. The results, absolute and relative errors are illustrated in some tables and diagrams. In addition, a comparison is made between the absolute errors obtained by the proposed method and two other methods; one uses a hybrid approach and the other applies second Chebyshev wavelet.

An Intelligent Feedforward Controller Utilizing a Modified Gorilla Troops Optimization for Nonlinear Systems

This paper presents an intelligent feedforward controller based on the feedback linearization approach to control nonlinear systems. In particular, the nonlinear autoregressive moving average (NARMA-L2) network is trained to reproduce the forward dynamics of the controlled system. Consequently, the trained NARMA-L2 network can be immediately integrated into the inverse feedforward control (IFC) structure. In order to improve the NARMA-L2 structure's ability to approximate nonlinear systems, the NARMA-L2 controller is comprised of two wavelet neural networks (WNNs). In addition, the RASP1 function was used as the mother wavelet function in the structure of the WNN rather than the more common Mexican Hat, Gaussian, and Morlet functions. To prevent the limitations of gradient descent (GD) methods, an artificial gorilla troops optimization (GTO) algorithm is used to determine the optimal settings for the NARMA-L2 inverse controller parameters. In particular, a modified version of the GTO algorithm, which is called the Modified GTO (MGTO) algorithm, is proposed in this work for training the NARMA-L2 inverse controller. This algorithm has demonstrated superior optimization outcomes in comparison to other methods. The effectiveness of the proposed control strategy is demonstrated using two nonlinear dynamical systems. Specifically, several evaluation tests are used to assess the effectiveness of the WNN-based NARMA-L2 in terms of control accuracy and robustness against external disturbances in each of the systems under consideration. These tests clearly demonstrated the effectiveness of the control system. Finally, a comparison study showed that the proposed WNN-based NARMA-L2 controller achieved better control results compared to the multilayer perceptron (MLP) and the radial basis function (RBF)-based NARMA-L2 controllers.

SCW-iterative-computational method for solving a wide class of nonlinear fractional optimal control problems with Caputo derivatives

The present paper offers an iterative-computational approach based on left and right operational matrices of the second Chebyshev wavelets (SCWs) in solving nonlinear fractional optimal control problems (FOCPs). The operational matrices are constructed by the hat functions (HFs). We first introduce necessary optimality conditions applying variational methods in a form of fractional two-point boundary value problem (TPBVP). For solving TPBVP, a fixed point iterative sequence, starting with an initial guess, in the hope of achieving more accurate solutions, is suggested. The SCWs-coefficients are easily obtained without any integrating. The order of convergence of the proposed method is accomplished in a new normed space. In the end of the article, some illustrative examples are provided to confirm the ability of the suggested technique.

A low-cost computational method for solving nonlinear fractional delay differential equations

The present work is devoted to proposing a low-cost spectral method based on the modified hat functions for solving fractional delay differential equations. The fractional derivative is considered in the sense of Caputo. In order to solve the considered problem, the existing functions in it are approximated using the basis functions. By employing some important properties of the basis functions, Caputo derivative and Riemann–Liouville fractional integral, the main problem is transformed into some systems of nonlinear algebraic equations including two unknown parameters. This procedure mainly simplifies the problem and gives its approximate solution after solving the resulting systems. In addition, the computational complexity of the derived system is investigated. An error analysis is discussed to show the convergence order of the method. Finally, the suggested technique is applied to some sample problems with the aim of checking its validity and accuracy.

A novel and efficient operational matrix for solving nonlinear stochastic differential equations driven by multi-fractional Gaussian noise

In this research study, we present an efficient method based on the generalized hat functions for solving nonlinear stochastic differential equations driven by the multi-fractional Gaussian noise. Based on the generalized hat functions, we derive a stochastic operational matrix of the integral operator with respect to the variable order fractional Brownian motion, for the first time so far. Also, we establish a procedure to generate the variable order fractional Brownian motion. Then, we use them to provide numerical solutions for the proposed problems. In addition, the convergence of the new method is theoretically analyzed. Moreover, we solve the stochastic logistic equation, stochastic population growth model, and three test problems to confirm the efficiency of the new method. The obtained results are compared with other existing methods used for solving these problems.

Numerical solution of nonlinear stochastic Itô–Volterra integral equation by stochastic modified hat function operational matrices

In this article, we present a numerical method to approximate for solving nonlinear Stochastic Itô–Volterra integral equations. This method is based on the modification of hat functions (MHFs) that introduce an operational matrix of integration. This reduces solving algebraic equations by using this approach and the collocation points. The researcher conducted a convergence analysis. Some numerical examples illustrate the accuracy and computational efficiency of the proposed methods.

B-splines on sparse grids for surrogates in uncertainty quantification

Robust prediction of the behavior of complex physical and engineering systems relies on approximating solutions in terms of physical and stochastic domains. For higher resolution and accuracy, simulation models must increase the number of deterministic and stochastic variables and therefore further increase the dimensionality of the problem. Sparse grids are an established technique to tackle higher-dimensional problems. Their efficient tensor product structure allows the creation of accurate surrogates from few model evaluations. Classical approaches use hat functions, resulting in non-differentiable surrogates, or global basis functions, resulting in potential instabilities. Therefore, we propose using modified not-a-knot B-splines to overcome both problems. Additionally, we use established spatially adaptive refinement criteria to reduce the number of model evaluations even further. We compare our technique to other data-driven uncertainty quantification methods in a real-world benchmark for probabilistic risk assessment for carbon dioxide storage in geological formations.

A numerical technique for solving nonlinear fractional stochastic integro-differential equations with n-dimensional Wiener process

This paper deals with the numerical solution of nonlinear fractional stochastic integro-differential equations with the n-dimensional Wiener process. A new computational method is employed to approximate the solution of the considered problem. This technique is based on the modified hat functions, the Caputo derivative and a suitable numerical integration rule. Error estimate of the method is investigated in detail. At the end, illustrative examples are included to demonstrate the validity and effectiveness of the presented approach.

Guaranteed and robust L2-norm a posteriori error estimates for 1D linear advection problems

We propose a reconstruction-based a posteriori error estimate for linear advection problems in one space dimension. In our framework, a stable variational ultra-weak formulation is adopted, and the equivalence of the L2-norm of the error with the dual graph norm of the residual is established. This dual norm is showed to be localizable over vertex-based patch subdomains of the computational domain under the condition of the orthogonality of the residual to the piecewise affine hat functions. We show that this condition is valid for some well-known numerical methods including continuous/discontinuous Petrov–Galerkin and discontinuous Galerkin methods. Consequently, a well-posed local problem on each patch is identified, which leads to a global conforming reconstruction of the discrete solution. We prove that this reconstruction provides a guaranteed upper bound on the L2 error. Moreover, up to a generic constant, it also gives local lower bounds on the L2 error, where the constant only depends on the mesh shape-regularity. This, in particular, leads to robustness of our estimates with respect to the polynomial degree. All the above properties are verified in a series of numerical experiments, additionally leading to asymptotic exactness. Motivated by these results, we finally propose a heuristic extension of our methodology to any space dimension, achieved by solving local least-squares problems on vertex-based patches. Though not anymore guaranteed, the resulting error indicator is still numerically robust with respect to both advection velocity and polynomial degree in our collection of two-dimensional test cases including discontinuous solutions aligned and not aligned with the computational mesh.

Numerical Solutions of Multi-order Fractional Antiperiodic Boundary Value Problems

In this paper, we propose the numerical scheme for solving nonlinear higher multi-order Caputo fractional differential equations with integral and antiperiodic boundary conditions, establish the existence of an approximate solution to our problem, and prove the convergence of our numerical method. The operational matrix of fractional integration based on hat functions is used to solve the given problem. Also, we present a numerical example to illustrate our main results.

Numerical solution of two dimensional stochastic Volterra–Fredholm integral equations via operational matrix method based on hat functions

Numerical solution of two-dimensional (2D) stochastic integral equations due to randomness has its own difficulties. For instance, most of them do not have analytical solution or obtaining their analytical solution is very hard. So, one of the essential requirement is presenting an efficient method to approximate the solutions of these equations with proper precision. In this paper, we introduce a numerical technique based on two dimensional modification of hat functions (2D MHFs) to estimate the numerical solution of 2D linear stochastic Volterra–Fredholm integral equations. In this approach, first the mentioned equation is transformed into a linear system of algebraic equations and then this system is solved via a direct or numerical method. Furthermore, it is proved that the order of convergence of the proposed method is $$O(h^3)$$ . Finally, accuracy of this scheme is measured by solving two test examples via described technique.

A new spectral method based on two classes of hat functions for solving systems of fractional differential equations and an application to respiratory syncytial virus infection

We propose a new spectral method, based on two classes of hat functions, for solving systems of fractional differential equations. The fractional derivative is considered in the Caputo sense. Properties of the basis functions, Caputo derivatives, and Riemann-Liouville fractional integrals, are used to reduce the main problem to a system of nonlinear algebraic equations. By analyzing in detail the resulting system, we show that the method needs few computational efforts. Two test problems are considered to illustrate the efficiency and accuracy of the proposed method. Finally, an application to a recent mathematical model in epidemiology is given, precisely to a system of fractional differential equations modeling the respiratory syncytial virus infection.

Solving Fractional Variational Problem Via an Orthonormal Function

In the present paper, a direct numerical technique for solving fractional optimal control problems based on an orthonormal wavelet, is introduced. First we approximate the involved functions by Sine-Cosine wavelet basis; then, an operational matrix is used to transform the given problem into a linear system of algebraic equations, which is easier. In fact operational matrix of Riemann-Liouville fractional integration and derivative of Sine-Cosine wavelet are employed to achieve a linear algebraic equation. The mentioned matrices are derived via hat functions. The solution of transformed system, gives us the solution of original problem. Two numerical examples are also given. Finally, the paper is ended with conclusion

A numerical approach for solving fractional optimal control problems using modified hat functions

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann–Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann–Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.