Search results

This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types).

The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations.

Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods.

The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

This paper sets out to introduce a numerical method to obtain solutions of Fredholm‐Volterra type linear integral equations.

The flow of the paper uses well‐known formulations, which are referenced at the end, and tries to construct a new approach for the numerical solutions of Fredholm‐Volterra type linear equations.

The approach and obtained method exhibit consummate efficiency in the numerical approximation to the solution. This fact is illustrated by means of examples and results are provided in tabular formats.

Although the method is suitable for linear equations, it may be possible to extend the approach to nonlinear, even to singular, equations which are the future objectives.

In many areas of mathematics, mathematical physics and engineering, integral equations arise and most of these equations are only solvable in terms of numerical methods. It is believed that the method is applicable to many problems in these areas such as loads in elastic plates, contact problems of two surfaces, and similar.

The paper is original in its contents, extends the available work on numerical methods in the solution of certain problems, and will prove useful in real‐life problems.

The purpose of this paper is to develop a numerical method based on quintic B‐spline to solve the linear and nonlinear Fredholm and Volterra integral equations.

The solution is collocated by quintic B‐spline and then the integral equation is approximated by the Gauss‐Kronrod‐Legendre quadrature formula.

The arising system of linear or nonlinear algebraic equations can solve the linear combination coefficients appearing in the representation of the solution in spline basic functions.

The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.

The paper provides new method to solve the linear and nonlinear Fredholm and Volterra integral equations.

This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

Based on the original methods of Adomian a decomposition method has been developed to find the analytic approximation of the linear and nonlinear Volterra‐Fredholm (V‐F) integro‐differential equations under the initial or boundary conditions.

Designed around the methods of Adomian and later researchers. The methodology to obtain numerical solutions of the V‐F integro‐differential equations is one whose essential features is its rapid convergence and high degree of accuracy which it approximates. This is achieved in only a few terms of its iterative scheme which is devised to avoid linearization, perturbation and any transformation in order to find solutions to given problems.

The scheme was shown to have many advantages over the traditional methods. In particular it provided discretization and provided an efficient numerical solution with high accuracy, minimal calculations as well as an avoidance of physical unrealistic assumptions.

A reliable method for obtaining approximate solutions of linear and nonlinear V‐F integro‐differential using the decomposition method which avoids the tedious work needed by traditional techniques has been developed. Exact solutions were easily obtained.

The new method had most of its symbolic and numerical computations performed using the Computer Algebra Systems‐Mathematica. Numerical results from selected examples were presented.

A new effective and accurate methodology has been developed and demonstrated.

Aims to solve Fredholm and Volterra non‐linear integral equations of the first kind. Uses the Adomian method, but since these equations are not under the canonical form u‐Nu = f, proposes some transformations for reducing the integral equations to integral equations of the second kind, much more appropriate. Uses a perturbation method for Fredholm equations. Concerning Volterra equations, uses a differentiation of the original equation, under sufficient regularity conditions, for obtaining a canonical form of Adomian.

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.

The results of this paper are theoretical and analytical in nature.

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.

The results are theoretical and analytical.

The results were applied to solving nonlinear integral equations.

The results has several social applications.

The results of this paper are new.

The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically.

For this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time.

By applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated.

Two numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

The purpose of this paper is to find methods for solving linear or non‐linear integral equations. The decomposition method may be applied, but a difficulty arises when it is applied to integral equations of the first kind which are not in canonical form. This paper aims to describe some techniques to obtain canonical form of Adomian decomposition method (ADM) for solving integral equations of the first kind.

Among various numerical methods for solving integral equations, the ADM has shown a good skill for determining the solutions as polynomial series forms. In this paper, some methods based on regularization, derivation or substitution techniques are presented to permit the application of ADM to integral equations of the first kind.

By using the three proposed methods the decomposition method can then be easily used to solve integral equations of the first kind. The three methods allow to obtain the canonical form and a solution to find using the ADM.

The three methods provide the solution by the application of the decomposition method of George Adomian as extended by other researchers. All are efficient: the derivation and substitution techniques giving interesting results for linear and non‐linear equations of Volterra type; the regularization technique is also efficient for the equations of Fredholm type, but difficulties could arise from the calculations of the required series.

The research provides three efficient methods for solving integral equations of first kind.

The three methods illustrate the powerful use of the decomposition techniques pioneered by Adomian and a result of their applications may be applied for solving of functional equations of any kind. This paper shows some methods for applying the Adomian techniques to integral equations of the first kind.

The purpose of this paper is to solve systems of linear Volterra integral equations of the first kind by the Adomian decomposition method (ADM). An elegant and reliable technique is outlined to find canonical form of ADM.

The approximate solution of systems of linear Volterra integral equations is calculated in the form of series with easily computable components. In this work, some methods based on substitution techniques are presented to permit the application of ADM to systems of integral equations of the first kind.

The approach developed in this work is valuable as a tool for scientists and applied mathematicians. It provides immediate and visible symbolic terms of analytical solution as well as its numerical approximate solution to systems of integral equations of the first kind, without linearization or discretization. The presented technique has many advantages over the traditional methods because it takes into account some systems of integral equations of first kind where the kernels present some singularities.

A reliable method for obtaining approximate solutions of linear systems of integral equations of the first kind using the ADM which avoids the tedious work needed by traditional techniques has been developed.

The research provides a new efficient method for solving systems of integral equations of first kind using ADM. The convergence result is investigated also and some numerical examples are given to illustrate the importance of the analysis presented.

The technique is both innovative and efficient, and an original approach for solving any kind of systems of linear Volterra integral equations of the first kind.