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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.

Abstract 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.

A class of tempered fractional differential equations with terminal value problems are investigated in this paper. Discretized collocation methods on piecewise polynomials spaces are proposed for solving these equations. Regularity results are constructed on weighted spaces and convergence order is studied. Several examples are supported the theoretical parts and compared with other methods. [ABSTRACT FROM AUTHOR]

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique’s validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021).

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. [ABSTRACT FROM AUTHOR]

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.

The derivative estimation problem is addressed in this paper by using Volterra integral 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 modelling error is characterised herein as well as the ISS property of the estimation 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. [ABSTRACT FROM AUTHOR]

A theoretical investigation is presented for the dynamic interaction of a rigid circular disc and its surrounding space, which is a trimaterial transversely isotropic full-space, and the disc is undergoing a prescribed horizontal harmonic vibration. Because the equations of motion form a system of coupled partial differential equations in a cylindrical coordinate system, a system of two scalar potential functions are used to decouple the equations of motion. This approach results in a set of two separated partial differential equations, which may be transformed to some ordinary differential equations by using Fourier series along the angular coordinate and the Hankel integral transformations with respect to the radial coordinate in a cylindrical coordinate system. After solving the ordinary differential equations, the unknown functions are determined by satisfying relaxed boundary conditions, which themselves are transformed to a set of four coupled integral equations. These coupled integral equations are reduced to two coupled Fredholm-Volterra integral equations of the second kind, which are numerically solved in this paper. The proposed solutions are applied for a transversely isotropic half-space, and the results showed the conformation of the existing solutions. [ABSTRACT FROM AUTHOR]

In this paper we study the continuous dependence and the differentiability with respect to the parameter λ ∈ [λ1, λ2] of the solution operator S : [λ1, λ2] → L2[a, b] for a mixed Fredholm-Volterra type integral equation. The main tool is the fiber Picard operators theorem (see [9], [8], [11], [3] and [2]).

In this paper we propose a simple numerical method for approximating solutions of Fredholm–Volterra integral equations of the second kind. The method is based on Picard iteration and uses a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. The paper concludes with numerical examples and a discussion of the approximations proposed. [ABSTRACT FROM AUTHOR]

In this paper, conclusions obtained from this two method are compared with by using successive approximations and variational iteration method for approximate solution of a second-order linear differential equation with retarded argument. [ABSTRACT FROM AUTHOR]

The purpose of this study is to present a method for solving high order linear Fredholm-Volterra integrodifferential equations in terms of Chebyshev polynomials under the mixed conditions. The method is based on the approximation by the truncated Chebyshev series. The higher order linear Fredholm-Volterra integrodifferential equations and the conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Chebyshev coefficients. Combining these matrix equations and then solving the system yields the Chebyshev coefficients of the solution function. Finally, the effectiveness of the method is illustrated in several numerical experiments and error analysis is performed. [ABSTRACT FROM AUTHOR]

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique's validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the Chebyshev collocation method has been used to solve the functional integral equation of the first and second kind. Also the numerical solution of the two dimensional Fredholm–Volterra integral equations (F–VIE) of the second kind is considered. The Chebyshev collocation method transforms any integral equation into a system of linear algebraic equations. In this method the Chebyshev expansion coefficients of the solution is obtained. Finally some examples show the accuracy of this method. [Copyright &y& Elsevier]

A method is used to solve the Fredholm–Volterra integral equation of the first kind in the space L2(Ω)×C(0,T),Ω=(x,y)∈Ω:√ of x2+y2&les;a, z=0 and T<∞.The kernel of the Fredholm integral term considered in the generalized potential form belongs to the class C([Ω]×[Ω]), while the kernel of the Volterra integral term is a positive and continuous function which belongs to the class C[0,T). Also in this work the solution of the Fredholm integral equation of the first and second kind with a generalized potential kernel is discussed. Many interesting cases are derived and established from the work. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]