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In this article, a new numerical approach is introduced for the numerical solution of a wide class of boundary value problems (BVPs). The underlying strategy of the algorithm is based on embedding an integral operator, defined in terms of Green’s function, into Ishikawa fixed point iteration scheme. The validity of the method is demonstrated by a number of examples that confirm the applicability and high efficiency of the method. The absolute error or residual error computations show that the current technique provides highly accurate approximations. [ABSTRACT FROM AUTHOR]

The aim of this article is to extend the Variational Iteration Method (VIM) for the proper treatment of boundary value problems (BVPs). This is achieved by introducing a modified correction functional, expressed in terms of two Lagrange multipliers, that takes both endpoints of the interval into consideration. The settings of the generalized VIM are given for three classes of BVPs. The numerical results of two examples confirm the validity and high accuracy of this novel generalization of the VIM. [ABSTRACT FROM AUTHOR]

This paper employs a generalized transformation to investigate Kudryashov's equation with four nonlinear terms in order to construct traveling wave solutions and optical solitons. The proposed unified ansätze framework is manipulated to derive analytical solutions, which are expressed as a combination of general solutions to simpler equations or systems of equations that are integrable or have unique known solutions. Using this technique, one can produce families of new exact solutions as well as solitary solutions. In addition to other model solutions, the scheme reveals the elliptic functions of Jacobi, Weierstrass, and Riccati. These are categorized as singular, bright, and dark optical solitons. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a patching scheme to solve numerically a class of boundary value problems. The strategy is being founded on a merger of two methods, a modified decomposition technique applied on a small interval near the singularity and a fourth order spline collocation technique which is used on the rest of the domain of the problem. The convergence of the proposed method is analyzed and a fourth-order rate of convergence is proved. Some examples are given to test the efficiency of the scheme and to verify the order of the rate of convergence. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods. [ABSTRACT FROM AUTHOR]

In this work, a new alternative uniformly convergent iterative scheme is presented and applied for the solution of an extended class of linear and nonlinear third order boundary value problems that arise in physical applications. The method is based on embedding Green’s functions into well-known fixed point iterations, including Picard’s and Krasnoselskii–Mann’s schemes. Convergence of the numerical method is proved by manipulating the contraction principle. The effectiveness of the proposed approach is established by implementing it on several numerical examples, including linear and nonlinear third order boundary value problems. The results show highly accurate approximations when compared to exact and existing numerical solutions. [ABSTRACT FROM AUTHOR]

The aim of this article is to introduce a novel approach based on embedding Green’s function into a tailored linear integral operator then apply a well-known fixed point iterative scheme, such as Picard’s and Mann’s, for the numerical solution of initial and boundary value problems (IVPs and BVPs). The strategy provides insight into the proper setting of the variational iteration method (VIM) for BVPs. A convergence analysis of the method is included. A number of examples consisting of second and third order BVPs are presented to demonstrate the efficiency and applicability of the scheme. [ABSTRACT FROM AUTHOR]

The purpose of this article is to search for and build exact solutions to the Gerdjikov–Ivanov equation. The approach's main thrust is to manipulate a tailored ansätze or transformation to construct solutions that are created from simpler integrable differential operators or basis sets of functions. The method generates families of new optical solitary solutions and traveling wave solutions for the Gerdjikov–Ivanov equation. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this article is to demonstrate that the variational iteration method “VIM” is in many instances a version of fixed point iteration methods such as Picard’s scheme. In a wide range of problems, the correction functional resulting from the VIM can be interpreted and/or formulated from well-known fixed point strategies using Green’s functions. A number of examples are included to assert the validity of our claim. The test problems include first and higher order initial value problems. [Copyright &y& Elsevier]

Abstract: The basic aim of this study is to introduce and describe a patching approach based on a novel combination of the variational iterative method (VIM) and adaptive cubic spline collocation scheme for the solution of a class of self-adjoint singularly perturbed second-order two-point boundary value problems that model various engineering problems. The domain of the problem is decomposed into two subintervals: the VIM is implemented in the vicinity of the boundary layer while in the outer region the resulting problem is tackled by applying an adaptive cubic spline collocation scheme (ASS), which comprises the use of mapping/transformation redistribution functions or constructed grading functions. Numerical results, computational comparisons, appropriate error measures and illustrations are provided to testify the convergence, efficiency and applicability of the method. Performance of this method is examined through test examples that reveal that the current approach converges to the exact solution rapidly and with accuracy and that the convergence is uniform across the domain. The proposed technique yields numerical solutions in very good agreement with and/or superior to existing exact and approximate solutions. [Copyright &y& Elsevier]

Abstract: The aim of the paper is to propose a generalized ansätze for constructing exact solutions to nonlinear ordinary differential equations. This unified transformation is manipulated to acquire analytical solutions that are general solutions of simpler linear or nonlinear systems of ordinary differential equations that are either integrable or possess special solutions. The method is implemented to obtain several families of traveling wave solutions for a class of nonlinear evolution equations and for higher order wave equations of KdV type (I). [Copyright &y& Elsevier]

Abstract: The aim of this article is to introduce a novel Laplace variational numerical scheme, based on the variational iteration method (VIM) and Laplace transform, for the solution of certain classes of linear and nonlinear differential equations. The strategy is outlined and then illustrated through a number of test examples. The results assert that this alternative approach yields accurate results, converges rapidly and handles impulse functions and the ones with discontinuities. [Copyright &y& Elsevier]

Abstract: A finite element approach, based on the cubic -spline collocation, is presented for the numerical solution of a class of singularly perturbed two-point boundary value problems that possesses a boundary layer at one or two end points. Due to the existence of a layer, the problem is handled using an adaptive spline collocation approach constructed over a non-uniform Shishkin-like mesh, defined via a carefully selected generating function. To tackle the case of nonlinearity, if it exists, an iterative scheme arising from Newton’s method is employed. The rate of convergence is verified to be of fourth-order and is calculated using the double-mesh principle. The efficiency and applicability of the method are demonstrated by applying it to a number of linear and nonlinear examples. The numerical solutions are compared with both analytical and other existing numerical solutions in the literature. The numerical results confirm that this method is superior when contrasted with other accessible approaches and yields more accurate solutions. [Copyright &y& Elsevier]

Abstract: A numerical method based on finite differences and spline collocation is proposed for solving a generalized parabolic problem subject to non-classical conditions. It is shown numerically that this scheme has second-order convergence. The stability conditions relating the time step size to mesh sizes were examined through numerical examples. It turns out that the stability of the scheme in certain cases is subjected to a stability condition and depends on the nature of the problem. A number of specific test problems are solved to assess the efficiency and performance of the method. The numerical results obtained indicate that the approach is viable and yields accurate solutions when compared with the existing closed-form solutions and generates results consistent with recent numerical approaches. [Copyright &y& Elsevier]

Abstract: A finite-element approach, based on cubic B-spline collocation, is presented for the numerical solution of Troesch’s problem. The method is used on both a uniform mesh and a piecewise-uniform Shishkin mesh, depending on the magnitude of the eigenvalues. This is due to the existence of a boundary layer at the right endpoint of the domain for relatively large eigenvalues. The problem is also solved using an adaptive spline collocation approach over a non-uniform mesh via exploiting an iterative scheme arising from Newton’s method. The convergence analysis is discussed and is shown to depend on the eigenvalues; in particular, the rate of convergence is calculated using the double-mesh principle. To demonstrate the efficiency of the method, a number of special cases are considered. The numerical solutions are compared with both the analytical solutions and other existing numerical solutions in the literature. It is observed that the results obtained by this method are quite satisfactory and accurate, and the method is applicable for a wide range of cases when contrasted with other available solutions. [Copyright &y& Elsevier]

Abstract: The aim of this work is to introduce and investigate a new class of log-biharmonic functions. Certain geometrically motivated properties and results concerning starlikeness, convexity and univalence of elements within this class versus the corresponding harmonic functions are obtained and discussed. In particular, we consider the Goodman–Saff conjecture and prove that the conjecture is true for the logarithms of functions belonging to this class. [Copyright &y& Elsevier]

Abstract: In the present paper, a collocation finite element approach based on cubic splines is presented for the numerical solution of a generalized wave equation subject to non-local conservation condition. The efficiency, accuracy and stability of the method are assessed by applying it to a number of test problems. The results are compared with the existing closed-form solutions; the scheme demonstrates that the numerical outcomes are reliable and quite accurate when contrasted with the analytical solutions and an existing numerical method. [ABSTRACT FROM AUTHOR]

Abstract: A new approach implementing a modified decomposition method in combination with the cubic B-spline collocation technique is introduced for the numerical solution of a class of singular boundary value problems arising in physiology. The domain of the problem is split into two subintervals; a modified decomposition procedure based on a special integral operator is implemented in the vicinity of the singular point and outside this domain the resulting boundary value problem is tackled by applying the B-spline scheme. Performance of this method is examined numerically; the examples reveal that the current approach converges to the exact solution rapidly and with accuracy. Results show that the method yields a numerical solution in very good agreement with the existing exact and approximate solutions. [Copyright &y& Elsevier]

Abstract: In this paper, a finite element collocation approach using cubic B-splines is employed for the numerical solution of a generalized form of the nonlinear Klein–Gordon equation. The efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine-Gordon equation. The numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels. [Copyright &y& Elsevier]

Abstract: In the present paper a numerical method, based on finite differences and spline collocation, is presented for the numerical solution of a generalized Fisher integro-differential equation. A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed-form difference scheme. A number of test examples are solved to assess the accuracy of the method. The numerical solutions obtained, indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods. Convergence and stability of the scheme have also been discussed. [Copyright &y& Elsevier]

Abstract: This paper presents a perturbation analysis study of the flow of an electrically conducting power-law fluid in the presence of a uniform transverse magnetic field over a stretching sheet. The perturbation solutions for small and large values of the mixed convection parameter are explored. The asymptotic behavior of the solutions was examined for different values of the power-law index and the magnetic parameter. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of Stokes flow in curved channels is solved by applying the matched eigenfunction expansion method. The flow region is partitioned into a union of sectorial subregions. Then, within each subregion, the stream function is represented by an expansion of Papkovich–Fadle eigenfunctions. The coefficients in these expansions are obtained by requiring weak continuity of the stream function across common interfaces. Biorthogonality conditions in cylindrical coordinates are utilized. [Copyright &y& Elsevier]

In this paper, a Laplace transform decomposition numerical algorithm is introduced for solving Bratu’s problem. The numerical scheme is based on the application of Laplace transform integral operator to the differential equation. The nonlinear term is then decomposed and an iterative algorithm is constructed for the determination of the infinite series solution. The technique is illustrated with two numerical examples and the results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme. [Copyright &y& Elsevier]

In the present paper, a numerical algorithm is implemented for the order verification of the asymptotic expansion solution of a second-order nonlinear differential equation arising in general relativity. The method is well-adapted and illustrated for a general class of nonlinear perturbation problems. The numerical scheme can be applied to asymptotic problems as long as the numerical solution is also available. [Copyright &y& Elsevier]

In the present paper, we derive the biorthogonality condition for Stokes flow in wedge-shaped trenches bounded by radial lines and concentric circles centered at the vertex of the wedge. This condition, is a property satisfied by the eigenfunctions and adjoint eigenfunctions needed to compute the coefficients of the eigenfunction expansion solution of the corresponding Stokes flow problem. [Copyright &y& Elsevier]

In this paper, a decomposition algorithm is presented for the solution of the coupled integral H-like equations of Chandrasekhar that arise in transport theory. An important feature of this numerical technique is its rapid convergence and the high degree of accuracy by which it approximates a solution using only few terms of its iterative scheme. [Copyright &y& Elsevier]

This work introduces two new sixth-order (3+1)-dimensional nonlinear Schrödinger equations with higher-order dispersive terms influenced by cubic–quintic–septic (CQS) nonlinearities. We formally retrieve bright and dark optical soliton solutions for each higher dimensional model. In addition, we find certain constraints on the parameters to determine these optical bright and dark solutions. Other solutions of distinct structures, such as singular solutions, are also formally determined. [ABSTRACT FROM AUTHOR]

• Novel ansätze is introduced for constructing solitons to nonlinear nonintegrable equations PDEs. • Method is successfully applied to the (2+1)-dimensional Caloger–Bogoyavlenskii–Schiff (CBS) equation. • Singular soliton solutions for the CBS equation are obtained. • Method can be applied to wide spectrum of mathematical physics nonlinear non-integrable PDEs. The aim of this article is to introduce a generalized ansätze for constructing soliton and traveling wave solutions to nonlinear nonintegrable equations partial differential equations arising in various fields of physics. The applicability of the approach is confirmed by employing it to obtain families of exact solutions of the (2 + 1)-dimensional Caloger–Bogoyavlenskii–Schiff (CBS) equation. [ABSTRACT FROM AUTHOR]

A complex tanh-function method is presented for constructing optical soliton solutions for quadratic–cubic nonlinear medium. In addition, an ansätze is proposed for obtaining soliton solutions of the dimensionless form of the nonlinear Schrödinger's equation that governs the propagation of solitons through optical fibers. [ABSTRACT FROM AUTHOR]

The aim of the article is to apply an efficient semi-analytical algorithm for the approximate solution of a fractional differential equation that governs ray tracing through the crystalline lens. A Laplace decomposition method (LDM) is implemented for the solution of the nonlinear ray equation of fractional order that is complemented with initial conditions. This method is a combination of Laplace transform and a decomposition method. The nonlinear term is decomposed and a recursive algorithm is constructed for the determination of the infinite series solution. Numerical tests are conducted, and the results confirm that this strategy converges rapidly with high accuracy using only few iterates of the recursive scheme. [ABSTRACT FROM AUTHOR]