Order Integral Equations Research Articles

Characterization of the angular coefficient method on 2D and 3D piecewise smooth boundaries

The angular coefficient method represents a valid and efficient strategy to estimate the distribution of molecules in ultra-high vacuum systems. The problem is described through a Fredholm integral equation of the second kind that is usually solved with standard numerical methods, e.g., the finite element method or the Nyström quadrature method. In this work, we aim to rigorously study the underlying integral equation in order to verify some fundamental mathematical properties and justify the application and behaviour of such numerical methods. In particular, we address to the general scenario where domains are not globally smooth. In such context, boundary corners entail poorly regular integral kernels, which in turn lead to non-Lipschitz solutions and require the adoption of non-standard analysis techniques. By introducing the concept of vacuum-connection, we can establish a methodology to prove the well-posedness of the underlying problem. Furthermore, the undermined regularity of the analytical solution and the consequent lower numerical convergence rate are proved analytically and verified through simple numerical tests.

On the Boundary Functional of a Semi-Markov Process

In this paper, we consider the semi-Markov random walk process with negative drift, positive jumps. An integral equation for the Laplace transform of the conditional distribution of the boundary functional is obtained. In this work, we define the residence time of the system by generalized exponential distributions with different parameters via fractional order integral equation. The purpose of this paper is to reduce an integral equation for the Laplace transform of the conditional distribution of a boundary functional of the semi-Markov random walk processes to fractional order differential equation with constant coefficients.

Generalizations of Riemann–Liouville fractional integrals and applications

The notion of a generalized Riemann–Liouville fractional integral is introduced, and its domain, range, and properties are studied. The new notion and properties provide new insight and understanding into the classical Riemann–Liouville fractional integral and its properties. Based on the generalized Riemann–Liouville fractional integral, equivalences between linear first ‐order fractional differential equations (FDEs) and integral equations are established. These equivalence results are applied to obtain solutions of Abel type integral equations and to study the existence and uniqueness of generalized normal solutions of initial value problems for nonlinear first‐order FDEs.

Linear higher-order fractional differential and integral equations

We study the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L1(a,b) and C[a,b] when n≥ 2. We establish the equivalence in C[a,b] of the IVP of the nth order FDE subject to the initial condition u(i)(a)=ui for i in {0,1,...,n-1} when n≥2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n≥2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n≥2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory.

Approximate Solution of Boundary Value Problem for Heat Equation after Represented by Volterra Integral Equation of the First Kind

In this work, we study the regularization method for solving the Boundary Value Problem (BVP) for heat equation. The discretization method applied with two variables on Volterra integral equation in order to covert the problem into a linear operator equation after applied the separation of variables method to solve the partial differential equation. The regularization way used to obtain the estimate solution by using the Lavrentiev regularization method.

Nonexistence Results of Global Solutions for Fractional Order Integral Equations on the Heisenberg Group

Nonexistence Results of Global Solutions for Fractional Order Integral Equations on the Heisenberg Group

Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation

Abstract At present, various branches of mathematics have developed very mature, but they still lack an internal structural correlation. They are like “islands” in the universe. If we can find the universal conversion relationship of the three basic mathematical structures of geometry, arithmetic and logic, it will greatly expand the field of mathematics. The ultimate goal of mathematics in the past is to pursue its preciseness, while the ultimate goal of mathematics in the future is to pursue its unity. Fractional differential equation is a generalization of integer order calculus. It discusses the theory of any non integer order differential and integral equation. The rise of artificial intelligence technology just provides the most powerful laboratory for the unity of mathematics. To explore the mystery of the brain and achieve the ultimate goal of artificial intelligence technology, the key is to make a major breakthrough in basic mathematical research and find a basic theoretical system describing the mutual mapping and transformation of different mathematical paradigms. The unity of mathematics will undoubtedly become the forefront of the development of mathematics in the future, which will be the core of the research on the new generation of intelligent computer and artificial intelligence technology.

Solvability of Implicit Fractional Order Integral Equation in ℓ p 1 ≤ p < ∞ Space via Generalized Darbo’s Fixed Point Theorem

We present a generalization of Darbo’s fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations in ℓ p 1 ≤ p < ∞ space. The fundamental tool in the presentation of our proofs is the measure of noncompactness mnc approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations.

Analyzing the existence of solution of a fractional order integral equation: A fixed point approach

Abstract This paper signifies the beauty of fixed point theory in the context of attaining the sufficient condition for the existence of the solution of a non-linear functional-integral equation in an unbounded interval. Here a non-linear integral equation (NLIE) involving a fractional operator is taken in the form of an operator equation. Utilizing the perception of measure of noncompactness (MNC) along with certain relevant assumptions, it is proved that the operator equation satisfies the Darbo condition for the product of operators in a Banach algebra. Finally, the result obtained is verified by the assistance of the numerical example.

Improved fractional [formula omitted] controller for AVR system using Chaotic Black Widow algorithm

Optimal tuning of fractional order proportional integral derivative (FOPID) controller parameters for automatic voltage regulator (AVR) system is a complex problem which requires the solution of real order integral and differential equations. Therefore, a novel optimization technique called Chaotic Black Widow Optimization (ChBWO) algorithm is proposed to tune the parameters of the FOPID controller. A new cost function is defined with a combination of Zwe-Lee Gaing’s (ZLG) and integral of time multiplied absolute error (ITAE) objectives. The proposed controller performance (rise time, settling time, and overshoot) is compared with the existing state-of-the-art techniques. To verify the robustness of the proposed controller the plant parameters are deviated from -50% to 50%. The simulation results show that the proposed controller is robust to variations in plant parameters. Moreover, disturbance analysis is also carried out by incorporating sudden changes in the reference signal. The results demonstrate that the controller is stable, robust, and sustains the sudden disturbances.

On solution of generalized proportional fractional integral via a new fixed point theorem

The aim of this paper is the solvability of generalized proportional fractional(GPF) integral equation at Banach space mathbb{E}. Herein, we have established a new fixed point theorem which is then applied to the GPF integral equation in order to establish the existence of solution on the Banach space. At last, we have illustrated a genuine example that verified our theorem and gave a strong support to prove it.

Convolution Quadrature Time-Domain Boundary Element Method for Viscoelastic Wave Scattering by Many Cavities in a 3D Infinite Space

This paper presents a time-domain boundary element method (BEM) for viscoelastic wave scattering by many cavities in 3D infinite space. The convolution quadrature method (CQM) is applied to the convolutions of the time-domain boundary integral equation in order to improve the numerical stability of BEM. Scattering of an incident wave by cavities in a viscoelastic solid is solved by the developed time-domain BEM. The incident wave propagating in a viscoelastic solid in time-domain is obtained using the inverse Fourier transform of that in frequency-domain. In addition, the hybrid parallelization using OpenMP and MPI is used to save the computational time and memory.

Physics-informed Supervised Residual Learning for Electromagnetic Modeling

In this paper, we propose a non-stationary iterative physics-informed supervised residual learning scheme (NSIP-ISRL) as a general framework for modeling electromagnetic wave propagation in inhomogeneous medium. NSIPISRL is based on the residual neural network (ResNet) that maps the residuals of matrix equation to the update of solutions [1]. It incorporates the concept of non-stationary iterative method, in which physical principles are embedded in the solution process through matrix-vector multiplication. NSIPISRL is applied to solve 2D volume integral equations in order to model electromagnetic wave interaction with lossy scatterers. The results show that NSIPISRL has a good accuracy and a strong generalization ability. The trained network can be applied to various scenarios with different scatterers, different incident angles, and different frequencies and still maintain a good accuracy.

M‐Parameter Mittag–Leffler function, its various properties, and relation with fractional calculus operators

A number of Mittag–Leffler functions are defined in the literature which have many applications across various areas of physical, biological, and applied sciences and are used in solving problems of fractional order differential, integral, and difference equations. This paper aims to define the m‐parameter Mittag–Leffler function, which can be reduced to various already known extensions of the Mittag–Leffler function. Important properties like recurrence relations, differentiation formulae, and integral representations are discussed for the newly defined m‐parameter Mittag–Leffler function. Moreover, the new m‐parameter Mittag–Leffler function is expressed in terms of some well‐known special functions such as generalized hypergeometric function, Mellin‐Barnes integral, Wright generalized hypergeometric function, and Fox H‐function. The paper also deals with various integral transforms like Euler‐Beta, Whittaker, Laplace, and Mellin transforms of the m‐parameter Mittag–Leffler function. Fractional differential and integral operators are considered to highlight certain intriguing properties of the m‐parameter Mittag–Leffler function. We also use the above function to define a generalization of Prabhakar integral and discuss its properties. Further, the relation of the newly defined m‐parameter Mittag–Leffler function with various other functions such as exponential, trigonometric, hypergeometric, and algebraic functions is obtained and represented graphically using MATHEMATICA 12.

Solvability of generalized fractional order integral equations via measures of noncompactness

In this article, we work on the existence of solution of generalized fractional integral equations of two variables. To achieve our main objective, we establish a new fixed point theorem using measure of noncompactness and a new contraction operator which generalized the Darbo’s fixed point theorem (DFPT). Also we obtain the corresponding coupled fixed point theorem. Finally we apply this generalized DFPT on the generalized fractional integral equations of two variables and illustrate our findings with the help of an example.

Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations

Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations

A new general integral transform for solving integral equations

IntroductionIntegral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily.During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms. ObjectivesIn this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\\_transforms, Pourreza, Aboodh and etc. MethodsA new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation. ResultsIt is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems. ConclusionIt has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p(s) and q(s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform.

On Riemann-Liouville Fractional Calculus and F-Function

The objective of the work is to explore certain correspondences that prevail amongst Riemann-Liouville fractional calculus and the F-function. The work is expected to find utilization in areas of fundamental linear order fractional differential and integral equations involving the F-function.

Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations

This paper is devoted to the numerical scheme for a class of fractional order integrodifferential equations by reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials. Reproducing kernel function in the form of Jacobi polynomials is established for the first time. It is implemented as a reproducing kernel method. The numerical solutions obtained by taking the different values of parameter are compared; Schmidt orthogonalization process is avoided. It is proved that this method is feasible and accurate through some numerical examples.

Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations

In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Fréchet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Hölder condition; however, these operators verify the omega condition established. Then, the semilocal convergence balls are obtained and the R-order of convergence and error bounds can be obtained by following thee main theorem. Finally, we perform a numerical experience by solving a nonlinear Hammerstein integral equations in order to show the applicability of the theoretical results by obtaining the existence and uniqueness balls.