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The purpose of this paper is to investigate the analytical solution of a hyperbolic partial differential equation (PDE) and its application.

The change of variables and the method of successive approximations are introduced. The Volterra transformation and boundary control scheme are adopted in the analysis of the reaction-diffusion system.

A detailed and complete calculation process of the analytical solution of hyperbolic PDE (1)-(3) is given. Based on the Volterra transformation, a reaction-diffusion system is controlled by boundary control.

The introduced approach is interesting for the solution of hyperbolic PDE and boundary control of the reaction-diffusion system.

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 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.

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.

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.

The purpose of this paper is to discuss a numerical method for solving system of Volterra integral equations.

An expansion method known as Chebyshev collocation method is chosen to convert the system of integral equations to the linear algebraic system of equations, so by solving the linear algebraic system an approximate solution is concluded.

Some numerical results support the accuracy and efficiency of the stated method.

The paper presents a method for solving first and second kind system of integral equations.

The Adomian decomposition method is used and applied to the mathematical model of a biosensor. This model consists of a heat equation with non‐linear and non‐local boundary conditions. To obtain a canonical form of Adomian, an equivalent non‐linear Volterra integral equation with a weakly singular kernel is set up. In addition, the asymptotic behaviour of the solution as t → 0 and t → • by asymptotic decomposition is obtained. Finally, numerical results are given which support the theoretical results.

The purpose of this paper is to elevate the accuracy when predicting the gross domestic product (GDP) on research and development (R&D) and to develop the grey delay Lotka-Volterra model.

Considering the lag effects between input in R&D and output in GDP, this paper estimated the delay value via grey delay relation analysis. Taking the delay into original Lotka-Volterra model and combining with the thought of grey theory and grey transform, the authors proposed grey delay Lotka-Volterra model, estimated the parameter of model and gave the discrete time analytic expression.

Collecting the actual data of R&D and GDP in Wuhan China from 1995 until 2008, this paper figure out that the delay between R&D and GDP was 2.625 year and found the dealy time would would gradually be reduced with the economy increasing.

Constructing the grey delay Lotka-Volterra model via above data, this paper shown that the precision was satisfactory when fitting the data of R&D and GDP. Comparing the forecasts with the actual data of GDP in Wuhan from 2009 until 2012, the error was small.

The result shows that R&D and GDP would be both growing fast in future. Wuhan will become a city full of activity.

Considering the lag between R&D and GDP, this work estimated the delay value via a grey delay relation analysis and constructed a novel grey delay Lotka-Volterra model.

This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at high frequencies exhibits a fractional effect. In addition, for the fractional-order formulation, this paper aims to provide a more accurate numerical algorithm for solving the fractional differential equations.

This paper analyzes the frequency-dependence of soil electrical properties based on fractional calculus theory. A collocation method based on the Puiseux series is proposed to solve fractional differential equations.

The algorithm proposed in this paper can be used to solve fractional differential equations of arbitrary order, especially for 0.5th-order equations, obtaining accurate numerical solutions. Calculating the impact response of the grounding electrode based on the fractional calculus theory can obtain a more accurate result.

This paper proposes an algorithm for solving fractional differential equations of arbitrary order, especially for 0.5th-order equations. Using fractional calculus theory to analyze the frequency-dependent effect of soil electrical properties, provides a new idea for ground-related transient calculation.