In this paper, modified hat functions and improved hat functions are proposed to solve stochastic Ito ̂-Volterra integral equations with multi stochastic terms. A linear system of equations are achieved by replacing the vector and matrix coefficients and operational matrices in the equation which is easy to solve with mathematical softwares. Also, under some conditions the error of these methods are o(h^3) and o(h^4 ) . The accuracy and reliability of these two methods are studied by solving and comparing the answers with block pulse functions and hat functions.

In this paper, using a new method based on the generalized hat functions, we solve a class of fractional delay differential equations in which the fractional derivative is considered in the sense of Caputo. First, we introduce the generalized hat functions and their corresponding operational matrices. Then, in order to solve the considered problem, the existing functions are approximated using the basis functions. By employing the properties of generalized hat functions, the Caputo fractional derivative and the Riemann-Liouville fractional integral, a system of algebraic equations is obtained which by solving it, the unknown coefficients are determined. By substituting the resulting values, an approximation of the solution of the problem is obtained. In addition, the computational complexity of the resulting system is investigated. In continue, an error analysis of the method is given. Finally, the accuracy and efficiency of the proposed method are shown by presenting two examples.

In this paper, we propose a numerical method based on the generalized hat functions (GHFs) and improved hat functions (IHFs) to find numerical solutions for stochastic Volterra-Fredholm integral equation. To do so, all known and unknown functions are expanded in terms of basic functions and replaced in the original equation. The operational matrices of both basic functions are calculated and embeded in the equation to achieve a linear system of equations which give the expansion coefficients of the solution. We prove that the rate of the convergence is O(h2) and O(h4) for these two different bases under some conditions. Two examples are solved and the results are compared with those of block pulse functions method (BPFs) to show the accuracy and reliability of the methods.

A numerical method to solve nonlinear quadratic integral equations (QIE) is presented in this work. The method is based upon modification of hat functions (MHFs) and their operational matrices. By using this approach and the collocation points, solving the nonlinear QIE reduces to solve a nonlinear system of algebraic equations. The proposed method does not need any integration for obtaining the constant coefficients. Hence, it can be applied in a simple and fast technique. Convergence analysis and associated theorems are considered. Some numerical examples illustrate the accuracy and computational efficiency of the proposed method.

n this paper, ‎numerical solution of a system of fractional differential equations which is the model of an epidemic disease is considered‎. ‎To this aim‎, ‎first‎, ‎third degree hat functions and their properties are introduced‎. After that‎, ‎using the expansion of the existing functions in the system in terms of the basis functions‎, ‎the system under consideration is transformed to a system of algebraic equations that can be solved using iterative methods‎. Then‎, ‎by solving the problem with a given initial data and comparing the results with the reported real data‎, ‎efficiency of the method is shown‎.

In this article, two numerical approaches are presented to solve a system of three fractional differential equations that express the pollution of lakes.In our recent study, a new class of hat functions, called quasi-hat functions (QHFs), are constructed.The proposed approaches utilize modified hat functions (MHFs) and quasi-hat functions (QHFs).Fractional-order operational of MHFs and QHFs are used to build algorithms that transform the main problem into a system of six equations with six unknowns and three equations with three unknowns, respectively.Absolute errors of obtained approximate solutions and convergence analysis of the utilized approach will be studied.Finally, three examples are provided to illustrate the capabilities of these algorithms.The pollution monitoring results are reported in some tables and figures for different values of $\alpha$.

Introduction Optimal control problems occur in engineering, science and many other fields. An optimal control problem is a problem of optimization of an objective functional on a set of state and control variables, which is called the performance index, subject to dynamic constraints on the states and controls. In the case that the dynamic constraints include delay fractional differential equation, the problem is called a delay fractional optimal control problem. In this paper, we consider the following optimal control problem ....

The present article introduces an operational approach based on modified hat functions to solve the space-time-fractional differential equations in the Caputo sense. In this method, the derivative of the unknown function is considered as a linear combination of modified hat functions. We use the operational matrix of the Riemann–Liouville fractional integral of modified hat functions to approximate the Caputo fractional derivative in order to reduce the problem to a system of Sylvester equations. The error of the mentioned method is of the order O(h3). In addition, we examine several  numerical examples to confirm the ability of the proposed approach.

Introduction: A system of integral equations can describe different kind of problems in sciences and engineering. There are many different methods for numerical solution of linear and nonlinear system of integral equations. Material and methods: This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations. The proposed method reduced a system of integral equation to a system of algebraic equations that can be solved easily by known methods. Results and discussion: For showing the accuracy and capability of the proposed method, some numerical examples are proposed that their results compared by results of other methods, and shows the capability and the superiority of this method to other existed methods. Also this paper derived the computational cost and the error analysis of the proposed method. Conclusion: The following conclusions were drawn from this research. This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations. The proposed method reduced a system of integral equation to a system of algebraic equations that can be solved easily by known methods. The presented error analysis and solved problems show capability and the superiority of this method to other existed methods.

In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operational matrices of CHFs, the main problem is transformed into a number of trivariate polynomial equations. Error analysis and the convergence of the proposed method are evaluated, and the convergence rate is addressed. Ultimately, three examples are provided to illustrate the precision and capabilities of this algorithm. The numerical results are presented in some tables and figures.