Approximate solution of the system of nonlinear integral equation by Newton-Kantorovich method

The Newton-Kantorovich method is developed for solving the system of nonlinear integral equations. The existence and uniqueness of the solution are proved, and the rate of convergence of the approximate solution is established. Finally, numerical examples are provided to show the validity and the efficiency of the method presented.     

An efficient multistep iteration scheme for systems of nonlinear algebraic equations associated with integral equations

The aim of this research is to present a new iterative procedure in approximating nonlinear system of algebraic equations with applications in integral equations as well as partial differential equations (PDEs). The presented scheme consists of several steps to reach a high rate of convergence and also an improved index of efficiency. The theoretical parts are furnished, and several computational tests mainly arising from practical problems are given to manifest its applicability.     

On the mixed problem for quasilinear pseudoparabolic equation

In this paper the multidimensional mixed problem for the quasilinear pseudoparabolic equation u_t-L_xu-εL_xu_t=f(t,x,u) is considered. L_x is a differential operator, which composes (with boundary operator) a self adjoint operator. An existence, uniqueness and also continuous dependense on the small parameter ε>0 of generalized solution is proved. The estimation of the difference of exact and approximate solutions is obtained     

The global attractivity and asymptotic stability of solution of a nonlinear integral equation

In this paper, we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation. Some new results are given.     

Monotonic positive solution of a nonlinear quadratic functional integral equation

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equation have been studied in [4], [5], [6], [7] and [8]. Here we are concerning with a nonlinear quadratic integral equation of Volterra type and we shall prove the existence of at least one L₁-positive monotonic solution for that equation under Carathèodory condition.     

A numerical technique for solution of the MRLW equation using quartic B-splines

Numerical scheme based on quartic B-spline collocation method is designed for the numerical solution of modified regularized long wave (MRLW) equation. Unconditional stability is proved using Von-Neumann approach. Performance of the method is checked through numerical examples. Using error norms L₂ and L_∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the new method is established through comparison with the existing techniques.     

Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules

The Gauss product quadrature rules and collocation method are applied to reduce the second-kind nonlinear two-dimensional Fredholm integral equations (FIE) to a nonlinear system of equations. The convergence of the proposed numerical method is proved under certain conditions on the kernel of the integral equation. An iterative method for approximating the solution of the obtained nonlinear system is provided and its convergence is proved. Also, some numerical examples are presented to show the efficiency and accuracy of the proposed method.     

Exact solutions of the nonlinear Schrödinger equation by the first integral method

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the nonlinear Schrödinger equation.     

基于迭代正则化高斯-牛顿法的非线性Urysohn积分方程数值解

非线性Urysohn积分方程在许多领域中都有广泛的应用,但由于该方程具有不适定性的特点,数据的微小扰动可能导致解的巨大变化,给数值求解带来很大困难.为了获得稳定的、准确的数值解,本文利用迭代正则化高斯-牛顿法对此方程进行求解,给出了利用Sigmoid-型函数确定迭代正则化参数的方法.对一类重力测定问题进行了数值模拟,将得到的数值解和相应的精确解作比较.结果表明,本文提出的方法在求解非线性Urysohn积分方程时是可行的也是有效的.     

He's variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation

In this paper, we apply He''s Variational iteration method (VIM) for solving nonlinear Newell-Whitehead-Segel equation. By using this method three different cases of Newell-Whitehead-Segel equation have been discussed. Comparison of the obtained result with exact solutions shows that the method used is an effective and highly promising method for solving different cases of nonlinear Newell-Whitehead-Segel equation.     

On solvability of one nonlinear integral equation in the class of analytic functions

We consider a nonlinear integral equation of a special type that appears in the inverse spectral theory of integro-differential operators and whose unique solvability in the class of square-integrable functions is known. However, for some applied issues in order to construct effective algorithms for solving equations of this type, it is required to establish their solvability in the class of analytic functions. Assuming the free term of the nonlinear equation under consideration to be an entire function of exponential type, we prove that so is its solution. Leaning on this result we provide a constructive procedure for solving this equation in the class of square-integrable functions, which can be easily implemented numerically.     

On positive definite solution of a nonlinear matrix equation

In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A^*X^?αA = Q with α ∈ (0, ∞) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

A strong solution of a high-order mixed type partial differential equation with integral conditions

In this article, we study a mixed problem with integral boundary conditions for a high-order partial differential equation of mixed type. We prove the existence and uniqueness of a strong solution. The proof is based on energy inequality and on the density of the range of the operator generated by the considered problem.     

On a fractional integral equation of periodic functions involving Weyl-Riesz operator in Banach algebras

In this paper, we study the existence of periodic solutions for a nonlinear integral equation of periodic functions involving Weyl-Riesz fractional integral operator under the mixed generalized Lipschitz, Carathéodory and monotonicity conditions. The fixed point theorems due to Dhage are the main tool in carrying out our proofs.     

A new numerical approach for the solution of nonlinear Fredholm integral equations system of second kind by using Bernstein collocation method

This paper presents an efficient numerical method for finding solutions of the nonlinear Fredholm integral equations system of second kind based on Bernstein polynomials basis. The numerical results obtained by the present method have been compared with those obtained by B‐spline wavelet method. This proposed method reduces the system of integral equations to a system of algebraic equations that can be solved easily any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Galerkin solution of a singular integral equation with constant coefficients

Galerkin methods are used to approximate the singular integral equation

with solution φ having weak singularity at the endpoint −1, where a, b≠0 are constants. In this case φ is decomposed as φ(x)=(1−x)^α(1+x)^βu(x), where β=−α, 0<α<1. Jacobi polynomials are used in the discussions. Under the conditions fH^μ[−1,1] and k(t,x)H^μ,μ[−1,1]×[−1,1], 0<μ<1, the error estimate under a weighted L² norm is O(n^−μ). Under the strengthened conditions f^″H^μ[−1,1] and , 2α−<μ<1, the error estimate under maximum norm is proved to be O(n^2α−−μ+), where , >0 is a small enough constant.