An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution, which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations

In this paper, we prove the convergence of homotopy analysis method (HAM). We also apply the homotopy analysis method to obtain approximate analytical solutions of systems of the second kind Volterra integral equations. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method.     

Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

A spectral Jacobi-collocation approximation is proposed for Volterra delay integro-differential equations with weakly singular kernels. In this paper, we consider the special case that the underlying solutions of equations are sufficiently smooth. We provide a rigorous error analysis for the proposed method, which shows that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Finally, two numerical examples are presented to demonstrate our error analysis.     

Dynamic Risk Measures for Anticipated Backward Doubly Stochastic Volterra Integral Equations

Inspired by the consideration of some inside and future market information in financial market, a class of anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs) are introduced to induce dynamic risk measures for risk quantification. The theory, including the existence, uniqueness and a comparison theorem for ABDSVIEs, is provided. Finally, dynamic convex risk measures by ABDSVIEs are discussed.     

Collocation Methods for a Class of Volterra Integral Functional Equations with Multiple Proportional Delays

In this paper, we apply the collocation methods to a class of Volterra integral functional equations with multiple proportional delays (VIFEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solutions for this type of equations, and then analyze the convergence orders of the collocation solutions and give corresponding error estimates. The numerical results verify our theoretical analysis.     

Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions

The theory of a class of spectral methods is extended to Volterra integro-differential equations which contain a weakly singular kernel $(t-s)^{-\mu}$ with $0<\mu<1$. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in $L^\infty$-norm and weighted $L^2$-norm. The numerical examples are given to illustrate the theoretical results.     

An Approximate KAM-Renormalization-Group Scheme for Hamiltonian Systems

We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov–Arnold–Moser (KAM) theory and renormalization-group techniques. It makes the connection between the approximate renormalization procedure derived by Escande and Doveil and a systematic expansion of the transformation. In particular, we show that the two main approximations, consisting in keeping only the quadratic terms in the actions and the two main resonances, keep the essential information on the threshold of the breakup of invariant tori.     

An Expansion Approach for the Stationary Probability Distribution of Systems Driven by Colored Noise

The stationary probability distributiolr of the two-dimensional Fokker-Planck equation of systems driven by colored noise is obtained in terms of an expansion iteration. The solution is compared with that obtained by the one-dimensional effective Fokker-Planck equation approaches. The behavior of several leading terms in the expansion for both small and large correlation times is analyzed. Numerical results support the analytical approach.     

Singular and Hypersingular Integral Equations Techniques for Gyrotron Coaxial Resonators with a Corrugated Insert

For the first time rigorous theory is developed for eigen traveling TM modes in the resonator of the coaxial cavity gyrotron with a corrugated insert. This mathematical model can be applied for any corrugation parameters and wavelengths. Gyrotron simulation software is developed and allows to calculate mode eigenvalues, electromagnetic field components and Ohmic losses for eigen TE and TM modes. Results of the numerical investigations are presented for the ITER relevant 170 GHz coaxial cavity gyrotron developed in Forschungszentrum Karlsruhe, Germany.     

Diffraction Theory by Means of Singular Integral Equations. VIIUniform High-Frequency Asymptotics for the Diffraction of Plane Waves by a Slit

We consider the diffraction of plane waves by a slit between two semi-infinite halfplanes for arbitrary angles of incidence and DIRICHLET boundary condition (electric field strength parallel to the edges). By a function-theoretic technique, which treats the angles of observation and incidence on an equal footing, we derive a high-frequency asymptotic expansion for the far field amplitude, uniformly valid over the whole range of both angles. Our expansion in terms of products of generalized FRESNEL integrals is compared with the known limiting cases of normal and grazing incidence both analytically and numerically including a comparison with MATHIEU -series results.     

Diffraction Theory by Means of Singular Integral Equations VI Diffraction of Plane Waves by an Infinite Strip Grating

For an infinite plane grating formed by strips and gaps of equal width, a Dirichlet boundary value problem for the Helmholtz equation is solved rigorously by function-theoretic techniques. Plane wave excitation with an arbitrary angle of incidence is considered. The high frequency asymptotics of the solution is completely evaluated and compared with Kirchhoff's diffraction theory as well as with the asymptotics for a single strip. Extensive numerical data are laid down graphically.     

Convergence Analysis of Legendre-Collocation Methods for Nonlinear Volterra Type Integro Equations

A Legendre-collocation method is proposed to solve the nonlinear Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors in $L^2$-norm and $L^\infty$-norm will decay exponentially provided that the kernel function is sufficiently smooth. Numerical results are presented, which confirm the theoretical prediction of the exponential rate of convergence.     

On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal.     

Complex Tanh-Function Expansion Method and Exact Solutions to Two Systems of Nonlinear Wave Equations

The complex tanh-function expansion method was presented recently, and it can be applied to derive exact solutions to the Schrodinger-type nonlinear evolution equations directly without transformation. In this paper,the complex tanh-function expansion method is applied to derive the exact solutions to the general coupled nonlinear evolution equations. Zakharov system and a long-short-wave interaction system are considered as examples, and the new applications of the complex tanh-function expansion method are shown.     

Complex Tanh-Function Expansion Method and Exact Solutions to Two Systems of Nonlinear Wave Equations

The complex tanh-function expansion method was presented recently, and it can be applied to derive exact solutions to the Schroedinger-type nonlinear evolution equations directly without transformation. In this paper,the complex tanh-function expansion method is applied to derive the exact solutions to the general coupled nonlinear evolution equations. Zakharov system and a long-short-wave interaction system are considered as examples, and the new applications of the complex tanh-function expansion method are shown.     

New Predictor-Corrector Methods Based on Exponential Time Differencing for Systems of Nonlinear Differential Equations

We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams-Bashforth-Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.     

Approximate Analytical Solutions for a Class of Laminar Boundary-Layer Equations

A simple and efficient approximate analytical technique is presented to obtain solutions to a class of two-point boundary value similarity problems in fluid mechanics. This technique is based on the decomposition method which yields a genera/analytic solution in the form of a convergent infinite series with easily computable terms. Comparative study is carried out to show the accuracy and effectiveness of the technique.     

An Optical Carrier Generation Approach for Cellular Millimeter-Wave Radio Systems

An optical carrier generation approach is presented for cellular millimeter-wave radio systems. The well-known methods distribute the millimeter wave signal via fibers. They have drawbacks because they need very high-speed photonic compo nents that are expensive. This problem is overcome by the new approach utilizing an optically transmitted low-frequency reference signal to generate the millimeter wave carrier. The new approach applies inexpensive photonic components and is less sensitive to the fiber dispersion.     

An Optical Carrier Generation Approach for Cellular Millimeter-Wave Radio Systems

An optical carrier generation approach is presented for cellular millimeter-wave radio systems. The well-known methods distribute the millimeter wave signal via fibers. They have drawbacks because they need very high-speed photonic compo nents that are expensive. This problem is overcome by the new approach utilizing an optically transmitted low-frequency reference signal to generate the millimeter wave carrier. The new approach applies inexpensive photonic components and is less sensitive to the fiber dispersion.