A new and effective direct method to determine the numerical solution of linear and nonlinear differential-algebraic equations (DAEs) is proposed. The method consists of expanding the required approximate solution as the elements of CHEBYSHEV CARDINAL FUNCTIONS. The operational matrices for the integration and product of the CHEBYSHEV CARDINAL FUNCTIONS are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a differentialalgebraic equation can be transformed to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

In this study, an effective numerical method for solving fractional differential equations using CHEBYSHEV CARDINAL FUNCTIONS is presented. The fractional derivative is described in the Caputo sense. An operational matrix of fractional order integration is derived and is utilized to reduce the fractional differential equations to system of algebraic equations. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

In this paper, we intend to introduce the Sturm-Liouville fractional problem and solve it using the collocation method based on CHEBYSHEV CARDINAL polynomials. To this end, we first provide an introduction to the Sturm-Liouville fractional equation. Then the CHEBYSHEV CARDINAL FUNCTIONS are introduced along with some of their properties and the operational matrices of the derivative, fractional integral, and Caputo fractional derivative are obtained for it. Here, for the first time, we solve the equation using the operational matrix of the fractional derivative without converting it to the corresponding integral equation. In addition to efficiency and accuracy, the proposed method is simple and applicable. The convergence of the method is investigated, and an example is presented to show its accuracy and efficiency.

This study concentrated on the numerical solution of a nonlinear Volterra integral equation. The approach is accorded to a type of orthogonal wavelets named the CHEBYSHEV CARDINAL wavelets. The undetermined solution is extended concerning the CHEBYSHEV CARDINAL wavelets involving unknown coefficients. Hence, a system of nonlinear algebraic equations is drawn out by changing the introduced expansion to the predetermined problem, applying the generated operational matrix, and supposing the CARDINALity of the basis FUNCTIONS. Conclusively, the estimated solution is achieved by figuring out the mentioned system. Relatively, the convergence of the founded procedure process is reviewed in the Sobolev space. In addition, the results achieved from utilizing the method in some instances display the applicability and validity of the method.