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In this study, a new form of a quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve integral equations.

The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations of the second kind. As these equations arise in many applications, there is a constant need for accurate, but fast and simple to use numerical approximations to their solutions. The method proposed here uses successive approximations of the Mann type and a suitable cubature formula. Mann’s procedure is known to converge faster than the classical Picard iteration given by the contraction principle, thus yielding a better numerical method. The existence and uniqueness of the solution is derived under certain conditions. The convergence of the method is proved, and error estimates for the approximations obtained are given. At the end, several numerical examples are analyzed, showing the applicability of the proposed method and good approximation results. In the last section, concluding remarks and future research ideas are discussed.

In this work, a reliable and efficient numerical technique viz. the balancing collocation technique (BCT) has been introduced and employed to solve the linear two-dimensional Fredholm–Volterra integral (F–VI) equations. The technique reduces the solution of these integral equations to the solution of a linear system of algebraic equations. Furthermore, the convergence and error analysis have been presented for the proposed methodology. Apart from that, the existence and uniqueness of the solutions obtained have been discussed. In addition, certain numerical examples have been discussed to demonstrate the accuracy and effectiveness of the proposed technique. Furthermore, certain comparative graphs for the BCT solutions and exact solutions have been depicted to illustrate the efficiency of the technique.

The derivative estimation problem is addressed in this paper by using Volterra in- tegral operators which allow to obtain the estimates of the time-derivatives with fast convergence rate. A deadbeat state observer is used to provide the estimates of the derivatives with a given fixed-time convergence. The estimation bias caused by modeling error is characterized herein as well as the ISS property of the estima- tion error with respect to the measurement perturbation. A number of numerical examples are carried out to show the effectiveness of the proposed differentiator also including comparisons with some existing methods.

A method is used to solve the Fredholm-Volterra integral equation of the first kind in the spaceL2(Ω)×C(0,T),Ω={(x,y):x2+y2≤a},z=0, andT<∞. The kernel of the Fredholm integral term considered in the generalized potential form belongs to the classC([Ω]×[Ω]), while the kernel of Volterra integral term is a positive and continuous function that belongs to the classC[0,T]. Also in this work the solution of Fredholm integral equation of the second and first kind with a potential kernel is discussed. Many interesting cases are derived and established in the paper.