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This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili (KdV-KP) equation. This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation. The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric, hyperbolic, and rational solutions. The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations. Furthermore, the obtained solutions have not been reported in the previous literature and might have significant impact on future research.

A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E-functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f ( α ) is transcendental or algebraic if f ( z ) is an E-function and α ∈ Q ‾ ⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f ( z ) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f ( 0 ) ∈ Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6] .

The aim of this paper is to present an analytical study for the investigation of the reciprocal metamaterial gyro-bianisotropy effects on the input impedance of a dipole antenna. This study is based on the numerical resolution, using the spectral method of moments (SMoM), of the integral equation developed through the mathematical derivation of the appropriate spectral Green’s functions of the studied dipole configuration. The original obtained results are discussed and showed good agreements with the isotropic studied cases available in published literature. It is found that the imaginary valued magnetoelectric elements (chiral medium) exhibit an inverse effect on the resonant length compared to purely real elements (Tellegen medium), while negative elements decrease the input impedance. Combination of the two cases has led to more general complex media with further inspiring features.

This paper presents a study of the performance of the Tau method using orthogonal polynomials as basis functions for solving fourth-order partial differential equations with boundary conditions. Because of the good numerical stability properties of integral operators in compare to differential operator, we first convert this PDE problem to Volterra-Fredholm integral equation and then apply the numerical Tau method to solve the obtained integral equation. Applying the Tau method yields a system of the ordinary differential equation such that this system is solved by piecewise polynomial collocation methods. Convergence analysis and error estimation of the Tau method are discussed. The advantages of converting PDE to integral equation are shown by the numerical examples. For this aim, we consider two cases to solve the proposed examples. In case1, we apply the Tau method to solve the converted problem (Volterra-Fredholm integral equation) and in case 2, we solve PDE problem directly by Tau method. Comparing the numerical results, we observe that the obtained errors in case 1 are less than the errors in case 2.

The aim of this work is to give some fixed point results based on the technique of measure of noncompactness which extend the classical Darbo’s theorem. With the help of our Darbo-type theorem, we obtain the existence of solution of implicit fractional integral equations in C ( I , l p α ) (collection of all continuous functions from I = [ 0 , a ] ( a > 0 ) to l p α ), where l p α is a tempered sequence space. Finally, we present a numerical example to see the validity and practicability of our existence result.

The fuzzy integral equation is used to model many physical phenomena which arise in many fields like chemistry, physics, and biology, etc. In this article, we emphasize on mathematical modeling of the fuzzy fractional Fredholm–Volterra integral equation. The numerical solution of the fuzzy fractional Fredholm–Volterra equation is determined in which model contains fuzzy coefficients and fuzzy initial condition. First, an operational matrix of Chebyshev polynomial of Caputo type fractional fuzzy derivative is derived in fuzzy environment. The integral term is approximated by the Chebyshev spectral method and the differential term is approximated by the operational matrix. This method converted the given fuzzy fractional integral equation into algebraic equations which are fuzzy in nature. The desired numerical solution is to find out by solving these algebraic equations. The different particular cases of our model have been solved which depict the feasibility of our method. The error tables show the accuracy of the method. We also can see the accuracy of our method by 3D figures of exact and obtained numerical solutions. Hence, our method is suitable to deal with the fuzzy fractional Fredholm–Volterra equation.

Plasma generated by hypersonic aerocraft is one of the research subjects in computational electromagnetics. When the aerocraft flies at an ultra-high speed in the near space, plasma sheath will be generated in the surroundings and it will trigger varying changes in aerocraft radar electromagnetic scattering properties. Plasma sheath brings about tremendous impacts on the communication between the aerocraft and the outside world, and even ends with "blackout effects" to interrupt communication in severe cases. The interplay between the plasma sheath and the electromagnetic wave involves multiple disciplines covering aerodynamics, plasma sheath physics, and electromagnetic field theory. Therefore, it is challengeable to build a rigorous numerical modeling of the plasma sheath. In this paper, the latest technologies in the electromagnetic simulation of hypersonic aerocraft with plasma sheath are reviewed. Applied cases review and illustrate extensive approaches from approximate calculations to accurate calculations, including analytical methods, differential equation methods, integral equation methods and high-frequency approximation methods. The paper assesses the RCS of electromagnetic wave in analyzing the electromagnetic scattering problem in plasma. With the considerations the reality of electromagnetic simulation and future challenges, the paper talks about the efficiency, advantages and disadvantages of these approaches in simulating hypersonic aerocraft targets.

This paper presents a new method for simultaneous modeling the uncertainty of measurement data (necessary to define the boundary shape and boundary conditions) in boundary problems. The interval parametric integral equation system (interval PIES) was developed for solving boundary problems with input data defined in this way. The motivation for conducting this research was that this topic (simultaneous consideration of uncertainties of all input data) has appeared sporadically in the literature (mainly with uncertainly defined boundary conditions or other parameters). In this paper, the uncertainty was defined using interval numbers and modeled using interval arithmetic. The direct application of both classical and directed interval arithmetic caused the overestimation and obtained solutions were useless in practice. Therefore, modification of the directed interval arithmetic was developed. The reliability of the interval PIES solutions obtained using such arithmetic was verified on 2D problems described by Laplace’s equation. The solutions were compared with the interval analytical solutions (differently obtained), as well as with the solutions of exactly defined (without the uncertainty) numerical methods. All performed tests indicated the high potential of the method. Obtained interval solutions occurred to be less overestimated and not as time-consuming as presented alternative methods.

Ballistic Missile Trajectory Prediction (BMTP) is critical to air defense systems. Most Trajectory Prediction (TP) methods focus on the coast and reentry phases, in which the Ballistic Missile (BM) trajectories are modeled as ellipses or the state components are propagated by the dynamic integral equations on time scales. In contrast, the boost-phase TP is more challenging because there are many unknown forces acting on the BM in this phase. To tackle this difficult problem, a novel BMTP method by using Gaussian Processes (GPs) is proposed in this paper. In particular, the GP is employed to train the prediction error model of the boost-phase trajectory database, in which the error refers to the difference between the true BM state at the prediction moment and the integral extrapolation of the BM state. And the final BMTP is a combination of the dynamic equation based numerical integration and the GP-based prediction error. Since the trained GP aims to capture the relationship between the numerical integration and the unknown error, the modified BM state prediction is closer to the true one compared with the original TP. Furthermore, the GP is able to output the uncertainty information of the TP, which is of great significance for determining the warning range centered on the predicted BM state. Simulation results show that the proposed method effectively improves the BMTP accuracy during the boost phase and provides reliable uncertainty estimation boundaries.

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's integral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical problems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is different from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives.

The paper deals with an integral equation arising from a problem in mathematical biology. We propose approximating its solution by Nystrom methods based on Gaussian rules and on product integration rules according to the smoothness of the kernel function. In particular, when the latter is weakly singular we propose two Nystrom methods constructed by means of different product formulas. The first one is based on the Lagrange interpolation while the second one is based on discrete spline quasi-interpolants. The stability and the convergence of the proposed methods are proved in uniform spaces of continuous functions. Finally, some numerical tests showing the effectiveness of the methods and the sharpness of the obtained error estimates are given.

In this paper we analyze spectral collocation method on a special class of nonlinear Volterra Hammerstein integral equations of the third-kind. After explaining the method, convergence analysis is discussed and then some examples are considered to show the validity of the theoretical results.

Recently, a polygonal boundary element method (PBEM) has been developed for solving heat conduction problems. In the PBEM, the radial integration method (RIM) is employed to shift the domain and surface integrals in the boundary-domain integral equation into equivalent line integrals, and all the radial integrals are computed by Gaussian quadrature, which may obtain an accurate solution. However, the computation costs much. Therefore, analytical expressions of radial integrals are derived in this paper, concerning four kinds of varying thermal conductivities and two kinds of heat generation functions, aiming at improving the efficiency. Then all the radial integrals in the boundary-domain integral equation can be analytically calculated. Three examples are employed to examine the analytically-integrated PBEM for solving steady heat conduction problems. The results show that the proposed method is accurate, and it is more efficient than traditional PBEM.

The novelty of this paper is to derive a mild solution by means of recently defined Mittag-Leffler type functions of fractional stochastic Langevin equations of orders α ∈ ( 1 , 2 ] and β ∈ ( 0 , 1 ] whose coefficients satisfy standard Lipschitz and linear growth conditions. Then, we prove existence and uniqueness results of mild solution and show the coincidence between the notion of mild solution and integral equation. For this class of system, we construct fractional Euler–Maruyama method and establish new results on strong convergence of this method for fractional stochastic Langevin equations. We also introduce a general form of the nonlinear fractional stochastic Langevin equation and derive a general mild solution. Finally, the numerical examples are illustrated to verify the main theory.

The paper presents the approach for solving 2D elastoplastic problems with singular stress/strain fields by the parametric integral equation system (PIES) method. PIES is applied since it does not require the discretization of the plastic zone into elements, and instead, it uses globally declared parametric surface patches. The properties of such surfaces (e.g. the unit square is their domain) are used for automatically concentrating the interpolation nodes around the regions with singular stress/strain fields. This, in turn, limits the global number of nodes in favor of their accumulation in specific places. The proposed approach of node concentration requires only a few parameters to be defined. In order to take into account the complex nature of the plastic strain fields, a special method of their interpolation is selected. The Kriging approach effectively involves an interactive investigation of the spatial behavior of the strains to choose the best estimate. It is also the method that bases on irregularly spaced interpolation nodes. Some examples are solved, and obtained results are compared with the finite element method (FEM).

In the present paper, a mathematical modeling of effective properties of fully coupled thermo-electro-mechanical heterogeneous materials is elaborated. Constitutive equations combined with equilibrium equations lead to a coupled partial differential system. A methodological procedure is elaborated based on thermo-electro-elastic Green’s functions, integral equations, and a micro-macro approach. Localization tensors are derived using the Mori–Tanaka mean-field assumptions and the fully coupled effective behavior is obtained through averaging techniques. Due to the large dispersion between elastic, dielectric, and piezoelectric coefficients ill-conditioned localization tensors are resulted. Block matrix decomposition is elaborated to overcome this drawback and the so-regularized localization tensor’s inverses are used to get a well conditioned problem. Numerical computations are done in the general case of ellipsoidal inclusions and anisotropic behavior. Effective thermal conductivity and heat capacity coefficients are obtained with and without the heat strain effect for various volume fractions and types of inclusions. The global properties of thermo-electro-elastic reinforced composites are presented with respect to shape and volume fractions of inclusions.

Diffraction–radiation of regular waves by a body such as an offshore structure or a moored ship is considered within the classical framework of potential-flow theory, which is commonly used because it is realistic and practical. A simple analysis that is based solely on straightforward applications of Green’s classical identity to a ‘free-waterplane’ flow-model and a ‘rigid-waterplane’ flow-model is expounded. These alternative flow-models and the related straightforward applications of Green’s identity yield an elementary yet rigorous justification of the practical and effective method, called ‘combined boundary integral equation method’ (CBIEM), of Lau and Hearn for preventing irregular frequencies. This method is shown to be significantly more efficient, and arguably simpler and more logical, than the widely-used extended boundary integral method of Lee and Newman. A modification of the ‘combined boundary integral equation method’ is also given. This modification involves a pair of boundary integral equations that are mathematically equivalent to the pair of integral equations solved in the CBIEM, but are weakly singular.

The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with multiple decorations. We also construct free commutative matching Rota-Baxter algebras in a relative context, to emulate the action of the integral operators on the coefficient functions in an integral equation. We finally show that free commutative matching Rota-Baxter algebras give the free matching Zinbiel algebra, generalizing the characterization of the shuffle product algebra as the free Zinbiel algebra obtained by Loday.

The development of a formulation of the Boundary Element Method (BEM) for application to elastic-viscoplastic analysis of plate bending is presented in this work. Reissner's theory is used, valid for thin and thick plates. Initially, Reissner's plate bending theory is presented, including the consideration of physical nonlinearity, and then, integral equations applied to Reissner's plates are shown. The theory for considering viscoplasticity is presented, as well as the procedures for its use with the BEM. In order to solve the equations that govern the problem, an incremental procedure is used and von Mises’ and Tresca's yield criteria are considered. This process offers an alternative method of solution for elasto-plastic problems, when steady-state condition is reached. Numerical examples are presented and results are compared with available solutions.

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations ( fie s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems ( evp s) associated with the two-dimensional fie s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fie s in both L 2 and L ∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L 2 and L ∞ norms. The numerical illustrations are introduced for the error of these methods.

A particular case of the Hermite interpolation method, namely the two-point Taylor formula, is utilized to construct a numerical technique for solving Fredholm integral equations (FIEs) of the second kind. This method can be applied to approximate the solution of both linear and nonlinear FIEs, and systems of nonlinear FIEs. The sufficient conditions to guarantee the convergence of the proposed method are provided through our analytical studies. Also, the error estimation is presented for this method. Furthermore, the efficiency of the method is confirmed by applying it to solve several illustrative examples. Numerical experiments confirm that the method is easy to implement and gives accurate approximations in acceptable computational times.

Graphical abstract, Highlights • A new general integral transform which is covered all class of integral transform in the class of Laplace transform. • We investigated the application of this new transform for solving ODE with constant and variable coefficient. • This new transform can handle easily for fractional order integral equations and fractional order differential equations. • We have discussed the advantage and disadvantage of other integral transformed which is defined during last 2 decades. • We proved the related theorems for this new transform., Introduction Integral 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. Objectives In 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. Methods A 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. Results It 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. Conclusion It 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.

We show constructively that, under certain regularity assumptions, any system of coupled linear differential equations with variable coefficients can be tridiagonalized by a time-dependent Lanczos-like method. The proof we present formally establishes the convergence of the so-called ⁎-Lanczos algorithm and yields a full characterization of algorithmic breakdowns. From there, the solution of the original differential system is available in a finite and treatable number of scalar integral equations. This is a key piece in evaluating the elusive ordered exponential function both formally and numerically.

Process of enhancing testing-capacity regarding COVID-19 is a topic of interest. This task of enhancing is constrained by socio-economic background of a country either in favorable or unfavorable ways. In this paper, we investigate timing of enhancing testing-capacity as an optimal problem, where the enhancement is quantified via number of tests as an instant measure and recovered portion as a long-term measure. The proposed work is structured analogous to an optimal machine replacement model based on a non-linear integral equation. Overall model is partially identifiable and compatible parameter estimations are carried out for a specific case study covering an early stage scenario. In addition, scenario development criteria on demand and effort for enhancing testing-capacity are introduced for predictions. In one numerical experiment, it is observed that frequency of enhancing testing-capacity starts decreasing after two increments indicating a favorable direction amidst effort constraints.

By employing the kernel functions with the form of piecewise polynomials in the Sobolev reproducing kernel Hilbert spaces (RKHSs), a globally superconvergent numerical technique is proposed to solve the second kind linear integral equations of Fredholm type. This method has an order of global convergence O ( h 4 ) and O ( h 6 ) based on the kernel functions in the Sobolev RKHSs H 1 and H 2 , respectively. Three linear Fredholm integral equations, one Volterra-Fredholm integral equation and one nonlinear Fredholm integral equation are numerically solved by the present approach to verify the superconvergence and effectiveness.

In this work, nonlocal strain gradient integral model is applied to study the elastic buckling and free vibration response of functionally graded (FG) Timoshenko beam which is made of two constituents varying along thickness direction. The differential governing equations and corresponding boundary conditions are derived via the Hamilton's principle, and the relations between nonlocal stresses and strains are expressed as integral equations. The Laplace transform technique is applied to solve directly the integro-differential equations, and the explicit expression for bending deflections and moments, as well as cross-sectional rotation and shear force is expressed with eight unknown constants. The nonlinear characteristic equations to determine the characteristic buckling load and vibration frequency are derived explicitly in consideration of corresponding boundary conditions and constitutive constraints. The buckling load and vibration frequency from present model are validated against to the existed results in literature. Consistent softening and toughening responses with respect to length-scale parameters can be obtained.

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

This study proposes a novel estimation-based approach to solving asset pricing models for both stationary and time-varying observations. Our method is robust to misspecification errors while inheriting a closed-form solution. By representing the Euler equation into a well-posed integral equation of the second kind, we propose a penalized two-stage nonparametric estimation method and establish its optimal convergence under mild conditions. With the merit of penalized splines, our estimate is less sensitive to the spline setting and we also design a fast data-driven algorithm to effectively tune the key smoother, i.e. the penalty amount. Our approach exhibits excellent finite sample performance. Using the US data from 1947 to 2017, we reinvestigate the return predictability and find that the estimated implied dividend yield significantly predicts lower future cash flows and higher interest rates at short horizons.

The boundary integral equations (BIE) and displacement discontinuity methods (DDM) are formulated for solution of smooth curved crack problems within 2D elasticity and Reissner's plate bending theory. The equivalence between the direct boundary element method and the indirect boundary element method is shown here. The Chebyshev polynomials of the second kind are employed to solve the integral equations numerically. This enables determination of the stress intensity factors at the crack tips directly5 by the coefficients of Chebyshev polynomials. In the DDM, the coefficients of stress influence for the constant displacement discontinuity element are derived in closed form for the Reissner's plate bending problem. The degree of convergence of BIE is demonstrated with increasing numbers of the collocation points. Comparison is made with the analytical solutions from the stress intensity factor handbook for an embedded circular crack in a flat plate under tensile force. High accuracy of the presented numerical solutions makes possible to use them as benchmarks to evaluate the degree of accuracy for any numerical algorithms.

In this work, we aim at the study of the statistical and thermal properties of the oil-in-water Pickering emulsions which are stabilized by strongly adsorbed charged nanoparticles on the surfaces of their droplets. In this study, we have adopted a pair-potential of Sogami-Ise type which is the sum of a repulsive part and an attractive one; and depends, in particular, on the salt-concentration. In this work, we were interested in two kinds of problems, namely (1) the determination of the structural and thermal properties of Pickering emulsions, and (2) the investigation of the scaling of cluster aggregation of the oil-droplets, as the salt-concentration is varied, keeping the other pertinent factors (surface-charge of oil-droplets, their size and density and bath temperature) fixed. For the two problems, we regarded the oil-droplets as monodisperse small spheres assimilated to charged soft-colloids. First, we have computed all of the structural properties (through the radial-distribution-function) versus the salt-concentration using the Integral-Equation Theory with the Hybridized-Mean-Spherical Approximation. Second, using the same numerical appoach, we have determined the thermal properties (through pressure, internal energy and isothermal compressibility). Finally, we have demonstraded that the oil-droplets aggregate has a fractal structure, with the help of an elaborated scaling model which was found to be in good agreement with Integral-Equation Theory.

In this paper we consider a fractional spectral collocation method for solving linear multi-term fractional differential equations involving Caputo-type fractional derivative. By using integral equation method, the boundary value problem of the fractional differential equation is reformulated as an equivalent Volterra-Fredholm integral equation with weakly singular kernels. Then we apply the fractional spectral collocation method to solve the resulting equation. We ensure that the approximate equation has a unique solution and the approximate solution arrives at the optimal convergence order in the infinite norm. At last, numerical examples are given to confirm the theoretical results.

Summary In order to investigate size-dependent creeping plane microfluidic flow, a boundary element method is implemented that involves the calculation of interior quantities and multi-domain problems. The governing equations are formulated using the skew-symmetric character of the couple-stress tensor and its energy conjugate mean-curvature tensor, as established in consistent couple stress theory. This theoretical formulation includes one characteristic material length scale parameter l that represents the size-dependency of the problem. Here, we present the boundary integral representation and numerical implementation for two-dimensional size-dependent steady state creeping incompressible flow, in which velocities, angular velocities, force- and couple-tractions are the primary variables. Beyond the previously known singular fundamental solution kernels for point force and point couple in the velocity and angular velocity integral equations, here we present the integral equations for calculating internal mean curvatures and couple-stresses, along with their corresponding kernels. The formulation is then applied to solve some computational problems both to investigate the size effects on the response resulting from the theory, and to validate the strength of the numerical method. For this purpose, we study the important effect of different boundary conditions on the flow pattern and consider a multi-fluid domain Couette flow problem.

This paper deals with the solution of system of 2D-fuzzy Fredholm integral equations (2D-FFIEs) depend upon the parametric form fuzzy number; using an efficient algorithm called Optimal Homotopy Asymptotic Method (OHAM). The efficiency and effectiveness of the proposed technique is tested upon some numerical example and the results are compared with modified homotopy perturbation method, 2D triangular function method and Lagender interpolation. It is observed from the results that the suggested method is accurate, straightforward and convenient to solve the 2D-fuzzy Fredholm integral equations.

The paper presents the problem of modeling the uncertainty of the boundary shape in boundary problems (described by Laplace’s equation) and proposes a method for solving so-defined problems. The uncertainty of the boundary shape is modeled using interval numbers. The interval coordinates of control points, defined using classical and directed interval arithmetic, are considered in this paper. However, because of some disadvantages of well-known interval arithmetic, the authors propose a modification of directed interval arithmetic. This arithmetic is also used in the proposed modification of the traditional parametric integral equation system (PIES) method (previously used for exactly defined problems) to solve so-defined problems. Control points of the appropriate curves, necessary to define the boundary shape, are directly included in the mathematical formalism of the mentioned method. The developed interval method is tested on problems of various shapes, modeled with linear and curvilinear segments. The correctness of the obtained solutions is verified using proposed alternative methods. The obtained solutions indicate a very high potential of the proposed method in solving problems with an uncertainly defined boundary shape.

The new system of hypersingular integral equations (HSIEs) for the thermally insulated inclined cracks and thermally insulated circular arc cracks subjected to remote shear stress in bonded dissimilar materials was formulated by using the modified complex potentials (MCPs) function method with the continuity conditions of the resultant force, displacement and heat conduction functions. This new system of HSIEs is derived from the elasticity problem and heat conduction problem by using crack opening displacement (COD) function and temperature jump along the crack faces. The appropriate quadrature formulas were used to solve numerically the new system of HSIEs for the unknown COD function and the known traction along the crack as the right hand term. Numerical solutions for the value of nondimensional stress intensity factors (SIFs) at all the cracks tips are illustrated. The comparison of nondimensional SIFs for the cracks with and without thermal is also illustrated.

The main purpose of this paper is to construct and analyze a spectral collocation method for solving a general class of nonlinear systems of multi-dimensional integral equations. In order to obtain high-order accuracy for the approximation, the integral terms in the resulting equation are approximated by using the Legendre spectral quadrature rule. The spectral rate of convergence for the proposed method is established in the L 2 -norm showing that the error of approximate solution decays exponentially. Numerical examples are presented to confirm this theoretical prediction.

For the two-dimensional elasticity problems, we give a uniqueness and existence analysis via the single-layer potential approach leading to a system of integral equation that contains a weakly singular operator. For its numerical solutions we describe a O ( h 3 ) order quadrature method based on the specific integral formula including convergence and stability analysis. Moreover, the asymptotic expansion of errors with odd power O ( h 3 ) is got and the accuracy of numerical approximations can be improved to the order of O ( h 5 ) by the Richardson extrapolation algorithm (EA) once. The efficiency of the method is illustrated by two numerical examples.

Hyper-singular integral equations are often applied in the frequency-domain wave diffraction/radiation analyses of marine structures with thin plates or shell sub-structures. Their numerical solutions require the higher-order derivatives of the free-surface Green function featuring hyper-singularity, and hence the corresponding evaluation is very challenging. To circumvent the associated numerical difficulties, this paper will propose alternative formulations for the higher-order derivatives of both free-surface and Rankine-source parts of the Green function. For the free-surface term G F , the higher-order derivatives are analytically expressed by a combination of G F itself and its first-order horizontal radial derivative. Further, we derive an asymptotic representation, enabling us to deal exactly with a removable singularity in this representation. The superiority of the proposed formulation is demonstrated by comparing with a conventional direct differentiation. For the Rankine-source term, analytical expressions for the velocities induced by a uniform dipole distribution over a flat panel (involving second derivatives of the Rankine source term) are presented, which is directly relevant to numerical implementation based on constant panel methods. As illustrative examples, linear hydrodynamic coefficients of submerged circular impermeable and perforated plates are calculated for verification purposes. The proposed formulas are simple and easy to implement in the hyper-singular integral equations.

In this paper, we characterize the logarithmic singularities arising in the method of moments from the Green’s function in integrals over the test domain, and we use two approaches for designing geometrically symmetric quadrature rules to integrate these singular integrands. These rules exhibit better convergence properties than quadrature rules for polynomials and, in general, lead to better accuracy with a lower number of quadrature points. We demonstrate their effectiveness for several examples encountered in both the scalar and vector potentials of the electric-field integral equation (singular, near-singular, and far interactions) as compared to the commonly employed polynomial scheme and the double Ma–Rokhlin–Wandzura (DMRW) rules, whose sample points are located asymmetrically within triangles.

The isogeometric analysis boundary element method (IGABEM) has great potential in the simulation of heat conduction problems due to its exact geometric representation and good approximation properties. In this paper, the radial integration IGABEM (RI-IGABEM) is proposed to solve isotropic heat conduction problems in inhomogeneous media with heat source. Similar to traditional BEM, the domain integrals cannot be avoided since the foundational solution for the Laplace equation is used to derive integral equation. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral. In addition, using a simple transformation method, the uniform potential method is successfully applied to solve the strongly singular integrals, and the Telles scheme and the element sub-division method are used to solve the weakly singular integrals in RI-IGABEM respectively. In order to validate the accuracy and convergence of the RI-IGABEM in the analysis of the single or multiple boundary heat conduction problems, several 2D and 3D numerical examples are used to discuss the influence of some factors, such as the number of applied points, the order of basis functions, and the position of internal applied points.

Electromagnetic wave scattering from a Perfect Electric Conductor (PEC) strip located at the interface of dielectric-Topological Insulator (TI) medium by using Kobayashi Potential (KP) method for Transverse Magnetic (TM) and Transverse Electric (TE) polarization is represented and discussed. Initially, diffracted fields equations are expressed by integral equations terms that contain unknown weighting functions. By choosing proper expression for weighting functions and exploiting Weber-Schafheitlin integrals properties, some boundary conditions satisfied automatically. The procedure leads to matrix equations which their size are chosen sufficiently, ergo the solution converges with rapid pace. Formulation, validation and convergence analysis of the proposed approach is represented and for the result, effect of structure’s parameters on Radar cross section (RCS) of the configuration is elucidated and portrayed.

In this paper, a new and efficient Dual Boundary Element Method (DBEM) named Dual Boundary-Interface Integral Equation Method (DBIIEM) is presented to solve crack problems in objects composed of arbitrary number of different materials. The conventional DBEM has been widely used in crack analysis and proved to be one of the most effective BEM for solving crack problems, however it cannot be directly employed to solve more complicated multi-medium crack problems. The presented DBIIEM can remedy this deficiency by dually collocating the displacement boundary-interface integral equation on one of the multi-medium crack surface and traction boundary-interface integral equation on the other. The newly derived displacement and traction boundary-interface integral equations are naturally fit for solving multi-medium elasticity problems, and independent each other, therefore they can be employed together in multi-medium crack analysis. The established DBIIEM inherits the advantage of single -region formulation of DBEM for solving more complicated crack problems. Complex stress intensity factors which characterize the oscillatory singularity of crack tips located at the interface crack are evaluated. Several numerical examples are given to verify correctness of the presented method.

In this paper, we are interested in an inverse problem for the biharmonic equation to recover the Robin coefficients on a non-accessible part of the boundary in simply connected planar domain from a measured Riquier-Neumann data on the accessible part of the boundary. Our work is based on a system of nonlinear and ill-posed integral equations which is solved by Tikhonov regularization method to complete missing Cauchy data. The minimization problem will be considered to solve the inverse problem, and recover the Robin coefficients on a known boundary curve. We show the feasibility of this method by numerical reconstructions.

In this paper, we develop a machine learning method with the Least Squares Support Vector Regression (LS-SVR) for the numerical solution of Fredholm integral equations. Two different approaches are proposed for training the network by using the shifted Legendre kernel, the collocation and Galerkin LS-SVR approaches. As with the standard LS-SVR for known dataset regression, the formulation of the method gives rise to an optimization problem. An equivalent system of algebraic equations is then derived and in linear cases discussed in terms of the sparsity of the matrices and computational efficiency. Finally, the method is carried out on some numerical examples, including nonlinear and multidimensional cases to show the accuracy and efficiency of the method.

Based on the coupled oscillatory characteristic of mode-1 and mode-2 near the interface crack tip, one kind of element for the relative displacements of the upper and lower surfaces of the axisymmetric interface crack is constructed. A universal numerical method for solving hypersingular integral equations of axisymmetric interface circular crack is established. For arbitrary normal and shear axisymmetric loadings, they are expanded into Taylor series, and the first five loadings are numerically calculated in a wide range of bimaterial constant e=0-0.15. The detailed numerical results of the stress intensity factors K1 and K2 are given in tabular form. Comparisons with the existing exact solutions show that the numerical results in this paper have high accuracy. For arbitrary axisymmetric loadings, the stress intensity factors can be determined from the tabular numerical results by using the superposition principle.

In this article, we analyse the convergence of an iterative method called Adomian decomposition method to solve the Fuzzy Volterra Integral Equations (FVIE) with time lag, the uniqueness and convergence of the method are equivalent. The concentration of the method has been proven and the theoretical results obtained have been tested with a numerical example. AMS2010 Mathematics Subject Classification: 34A07, 34A12, 78M30.

The wave field of an infinite elastic layer weakened by a cylindrical cavity is constructed. The ideal contact conditions are given on the upper layer’s face and bottom face is rigidly fixed. The normal dynamic tensile load is applied to a cylindrical cavity’s surface at the initial moment of time. The Laplace and finite sin- and cos- Fourier integral transforms are applied successively directly to axisymmetric equations of motion and to the boundary conditions, on the contrary to the traditional approaches, when integral transforms are applied to solutions’ representation through harmonic and biharmonic functions. This operation leads to a one-dimensional vector inhomogeneous boundary value problem with respect to unknown displacements’ transformations. The problem is solved using a matrix differential calculus, which leads to an integral equation solved with a method of orthogonal polynomials. The field of initial displacements is derived after application of inverse integral transforms. The case of the steady-state oscillations was investigated. The normal stress on the rigidly fixed face of the elastic layer is constructed and investigated depending on the mechanical and dynamic parameters. Formulas for determining the normal stress for large values ​​of natural vibration frequencies were constructed.

The large-deflection analysis of a tapered beam-column under transverse loadings and restrained axially is presented. The proposed method is based on the Differential Transform Method (DTM) but with improved efficiency. This is achieved by separately analyzing and incorporating the differential transform of the trigonometric functions s i n θ and c o s θ with θ s - = ∑ k = 0 m u k s - k being the solution to the problem making possible to work with series of order m = 20 without incurring into excessive computational effort and calculation time. The transformed equations convert the governing integro-differential equation of the problem into a polynomial equation with the coefficients u k found by recurrence with its solution much easier to determine. The accuracy of the improved DTM depends on the order m of the series becoming the “exact” solution to the differential integral equation as m tends to infinity. The minimum value of m required in the expansion depends on the applied loads on the beam-column, its support conditions and geometry. Two classic beam-columns are analyzed herein using the proposed method: 1) a simple-supported beam-column partially restrained axially subject to a uniformly distributed transverse load; and 2) a tapered cantilever beam-column subject to a concentrated transverse load at the free end. The obtained results are compared with those reported by other researchers with excellent agreement showing the validity and simplicity of the proposed method and its corresponding equations.

In this paper stability of numerical solution of Fredholm integral equation of the first kind is studied for radial basis kernels which possess positive Fourier transform. As a result, the equivalence relation between strong and weak forms of partial differential equations (PDEs) is proved for some special radial test functions. Also the stability of boundary elements method (BEM) is proved analytically for Laplace and Helmholtz equations by obtaining Fourier transform of singular fundamental solutions applied in BEM. Analytical result presented in this paper is an extension of stability idea of radial basis functions (RBFs) used to interpolate scattered data described by Wendland in [51] . Similar to the interpolation, it is proved here mathematically that integral operators which have radial kernels with positive Fourier transform are strictly positive definite. Thanks to the stability idea presented in [51] , a positive lower bound for eigenvalues of these integral operators is found here, explicitly.

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.