Two Dimensional Fredholm Integral Equation with Time

In this paper, the existence and uniqueness of solution of the Fredholm‐Volterra integral equation (F‐VIE), with a generalized singular kernel, are discussed and proved in the spaceL2(Ω)×C(0,T). The Fredholm integral term (FIT) is considered in position while the Volterra integral term (VIT) is considered in time. Using a numerical technique we have a system of Fredholm integral equations (SFIEs). This system of integral equations can be reduced to a linear algebraic system (LAS) of equations by using two different methods. These methods are: Toeplitz matrix method and Product Nyström method. A numerical examples are considered when the generalized kernel takes the following forms: Carleman function, logarithmic form, Cauchy kernel, and Hilbert kernel.

Systems of Volterra and Fredholm integral equations have attracted much concern in applied sciences. The systems of Fredholm integral equations appear in two kinds. The system of Fredholm integral equations of the first kind [1–5] reads $$\begin{gathered} {f_1}\left( x \right) = \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ {f_2}\left( x \right) = \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ \end{gathered} $$ (11.1) where the unknown functions u(x) and v(x) appear only under the integral sign, and a and b are constants. However, for systems of Fredholm integral equations of the second kind, the unknown functions u(x) and v(x) appear inside and outside the integral sign. The second kind is represented by the form $$\begin{gathered} u\left( x \right) = {f_1}\left( x \right) + \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ v\left( x \right) = {f_2}\left( x \right) + \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt. \hfill \\ \end{gathered} $$ (11.2) The systems of Fredholm integro-differential equations have also attracted a considerable size of interest. These systems are given by $$\begin{gathered} {u^{\left( i \right)}}\left( x \right) = {f_1}\left( x \right) + \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ {v^{\left( i \right)}}\left( x \right) = {f_2}\left( x \right) + \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ \end{gathered} $$ (11.3) where the initial conditions for the last system should be prescribed.

Systems of Fredholm integral equations are essential in engineering fields for modeling complex systems involving dynamic behavior, material properties, and interactions between different forces or fluids. In this research, the Taylor expansion approach is utilized to solve the systems of linear Fredholm integral equations (SLFIEs) of the second kind in a novel manner. By combining the m th-order Taylor polynomial of unknown functions at an arbitrary point and employing the repeated integration method; the given system of linear Fredholm integral equations is transformed into a new system of linear equations of unknown functions and their derivatives. Eventually, this new system is solved to obtain demanded m th-order approximate solutions. An error analysis is given as well as several numerical examples to demonstrate the efficiency, accuracy, and ability of the proposed method. Furthermore, this method leads always to the exact solution if the exact solution is a polynomial of degree less than or equal to m .

Convergence of approximate solution of system of Fredholm integral equations

In this paper the modification in successive approximation method is defined to solve a system of linear fredholm integral equations(SLFIE) of the 2nd kind with symmetric kernel, also applied Aitken method depending on successive approximation method to the solve SLFIE of the 2nd kind and the algorithm for the methods are suggested, efficiency of the methods is clear through solving some numerical examples, the results in tables (1,2,3 and 4) indicate the accuracy of the approachs.

Formulation of the problem in general. During radio relay communication, electromagnetic waves propagate along the earth's surface, and due to refraction, the polarization of signals can change, which leads to loss of signal power. This problematic situation can be solved by polarization adaptation of antenna systems built on the basis of lattice structures that convert signals with any polarization into a circle. Such antenna arrays are polarization-holographic antennas.Analysis of recent researches and publications. In the theory of antenna synthesis, closed translucent surfaces, methods of synthesis of directional properties of reflective antenna arrays, which take into account the presence of mutual communication between irradiators of any type, as well as models of electrodynamic structures for arrays of different shapes are considered. This approach has a generalized nature and requires the adaptation of mathematical methods and models to antenna systems of a specific design. The polarization-holographic antenna can be considered as a non-homogeneous translucent structure, the problem of diffraction of electromagnetic waves in which it is expedient to solve the Fredholm integral equations of the first kind.Presenting the main material. The mathematical formalization of the electrodynamic model of a non-homogeneous translucent body, which in its properties corresponds to the polarization-adaptive antenna array, is considered as the formulation and solution of the inverse electrodynamic problem connecting the primary electromagnetic field, surface current, and surface impedance. This surface impedance is the holographic kernel of the integral equation, which makes it possible to synthesize the impedance surface for circularly polarized waves. Diffraction of electromagnetic waves on a multilayer impedance body is described by a system of Fredholm integral equations of the first kind for different resonant wavelengths.Conclusion. The method of Fredholm integral equations of the first kind makes it possible to determine the parameters of the antenna array through the transformation of the primary electromagnetic field into a secondary one based on the principles of holography. The result of solving the integral equation is its holographic core, which corresponds to the transparent (reflector) of the antenna. A system of integral equations was obtained, which mathematically formalizes the electrodynamic model of a planar antenna array with several layers, taking into account the mutual influence of these layers.The perspectives of future researches. Further studies should be considered the description of the propagation process of electromagnetic waves, taking into account their multiple reflections, in a planar impedance body and the improvement of the matrix method for determining the transmission and reflection coefficients in such bodies.

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).

Integral equations are essential in numerous domains of practical mathematics. This article introduces a straightforward, precise, and efficient iterative technique for resolving one-dimensional Fredholm integral equations of the second class. The suggested numerical method relies on Legendre wavelet functions. Utilizing these wavelets, the integral equation system is converted into a duo of interconnected systems of algebraic matrix equations. A finite iterative approach is employed to resolve these systems and ascertain the coefficients that formulate the approximate numerical solutions of the unknown functions. A variety of examples are included to evaluate the proposed numerical approach.

The aim of this paper is to develop numerical methods for solving systems of integral equations with cardinal splines. The unknown functions are expressed as a linear combination of horizontal translations of certain cardinal spline functions with small compact supports. Then a simple system of equations on the coefficients is obtained for the system of integral equations. It is relatively straight forward to solve the system of unknowns and an approximation of the original solution with high accuracy is achieved. Several cardinal splines are used in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined and the convergence rate is investigated. Examples are given to demonstrate the benefits of the methods.

This study deals with introducing spline function to find the approximate solution of the system of nonlinear Fredholm integral equations of the second kind. The benefit of spline functions was demonstrated by presenting several examples.

Scattering of obliquely incident surface gravity waves by an array of partial flexible vertical porous wave barriers of varied configurations is studied in finite water depth and deep water cases. For the mathematical modeling purpose, linearized water wave theory is considered. Firstly, the problem associated with double vertical barriers is handled for a solution. To solve the problem, the associated BVP (boundary value problem) is transformed into a system of Fredholm integral equations of the second kind. By using suitable quadrature formulae, these integral equations are converted to a system of linear algebraic equations. These are solved to compute various important physical quantities associated with wave scattering. The energy identities for both finite water depth and deep water cases are derived. These derived energy identities are used to check the computational accuracy and to quantify the loss of wave energy by the barrier system. Subsequently, the study is extended to deal with wave scattering by multiple non‐identical unequally spaced barriers using wide‐spacing approximation and results for equally spaced identical barriers are obtained as a special case. The methodology is further generalized to deal with Bloch problem to incorporate a periodic array of infinite barriers. Occurrences of Bragg resonance in case of multiple barriers are analyzed. From the results associated with the linearized water wave theory, the second‐order mean wave forces are computed and compared with the first‐order wave forces. The phenomena of cloaking for wave scattering by multiple impervious barriers are discussed for a wide variety of wave and structural parameters. The methodology used in this present study can easily be generalized to solve similar problems that arise in mathematical physics.

Fredholm–Volterra integral equation with singular kernel

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel\Abstracteng{In this article the problems of the unique classical solvability and theconstruction of the solution of a nonlinear boundary value problem for a fifth orderpartial integro-differential equations with degenerate kernel are studied. Dirichletboundary conditions are specified with respect to the spatial variable. So, the Fourierseries method, based on the separation of variables is used. A countable system of~thesecond order ordinary integro-differential equations with degenerate kernel is obtained.The method of degenerate kernel is applied to this countable system of ordinaryintegro-differential equations. A system of~countable systems of algebraic equations isderived. Then the countable system of nonlinear Fredholm integral equations is obtained.Iteration process of solving this integral equation is constructed. Sufficient coefficientconditions of the unique solvability of the countable system of nonlinear integralequations are established for the regular values of parameter. In proof of uniquesolvability of the obtained countable system of nonlinear integral equations the method ofsuccessive approximations in combination with the contraction mapping method is used.In the proof of the convergence of Fourier series the Cauchy--Schwarz and Besselinequalities are applied. The smoothness of solution of the boundary value problemis also proved.

In this work, the Hammerstein integral equation (HIE), with a generalized singular kernel, is considered and solved numerically, using Product Trapezoidal rule, Toeplitz matrix method and Product Nystrom method. The existence and uniqueness solution, under certain conditions, are considered. Moreover, numerical results when the kernel takes a generalized logarithmic form, Carleman function and Cauchy kernel are investigated. Also the error, in each case, is estimated.

This paper considers the lift coefficient of an airfoil with nonzero prescribed thickness and camber, along with a prescribed porosity distribution in a steady incompressible flow. In similar prior work, considering a Darcy-type porosity condition on the airfoil surface provides a Fredholm integral equation that can be solved exactly for Hölder-continuous porosity distributions. However, the generated lift predicted by the model diverges from the experimental data for porous airfoils with large values of the porosity parameter. This indicates a fundamental characteristic is missing in the mathematical model. Consequently, in this paper, the (linear) Darcy porosity condition is replaced by the (nonlinear) Forchheimer porosity condition. The Forchheimer porosity condition is decomposed into linear sections and furnishes a modified system of Fredholm integral equations, enabling an approximate solution by superposition for Hölder-continuous porosity distributions. The solution is then verified against the SD7003 airfoil results, and the comparison shows better agreement than considering the Darcy porosity condition. The methodology and results presented here may be combined with previous work on the aeroacoustics of porous airfoils with a Forchheimer boundary condition to address the conflicting aims of improving aerodynamic performance while reducing unwanted aeroacoustic emissions.