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The modified complex potentials (MCPs) functions are used to develop a new system of singular integral equations (SIEs) for a curvilinear crack in the upper part of bonded materials subjected to shear mode stress with the help of continuity conditions for resultant force and displacement functions. The unknown dislocation distribution function is mapped into a square root singularity function by using curved length coordinate method and the traction along the crack as the right hand term. The Gaussian quadrature rules were used to obtain the numerical solution for a new system of SIEs in order to compute the nondimensional stress intensity factors (SIFs) for these problems. Our results agree with those of the previous works. The findings have revealed that the nondimensional SIFs depend on the elastic constant ratio, crack geometries and the position of the cracks.

A new mathematical model is developed for the analytical study of two cracks in the upper plane of dissimilar materials under various mechanical loadings, i.e., shear, normal, tearing and mixed stresses with different geometry conditions. This problem is developed into a new mathematical model of hypersingular integral equations (HSIEs) by using the modified complex potentials (MCPs) function and the continuity conditions of the resultant force and displacement with the crack opening displacement (COD) function as the unknown. The newly obtained mathematical model of HSIEs are solved numerically by utilizing the appropriate quadrature formulas. Numerical computations and graphical demonstrations are carried out to observe the profound effect of the elastic constants ratio, mode of stresses and geometry conditions on the dimensionless stress intensity factors (SIFs) at the crack tips.

Triple circular arc cracks problems subjected to shear stress in half-plane elasticity is investigated. Modified complex potentials (MCP) with the free traction boundary condition are applied to formulate the hypersingular integral equation (HSIE) for the problems. The unknown crack opening displacements (COD) of the HSIE are solved numerically by using the appropriate quadrature formulas. Mode I and Mode II of nondimensional stress intensity factor (SIF) at all cracks tips are presented for the problems of three adjacent circular arc cracks, three circular arc cracks with dissimilar radius and three circular arc cracks in series in a half-plane. The results exhibit that as the crack opening angle increases and the distance of cracks closer to the boundary of half-plane, the nondimensional SIF increases. This indicates that the strength of material becomes weaker and the tendency of material to fail is higher.

A hypersingular integral equations (HSIEs) of the first kind on the interval [ 1 ; 1 ] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain is considered. Truncated series of Chebyshev polynomials of the third and fourth kinds are used to find semi bounded (unbounded on the left and bounded on the right and vice versa) solutions of HSIEs of first kind. Exact calculations of singular and hypersingular integrals with respect to Chebyshev polynomials of third and forth kind with corresponding weights allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature formula is extended for regular kernel integrals. Three examples are provided to verify the validity and accuracy of the proposed method. Numerical examples reveal that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights.

In this paper, a problem of anomalous solute transport in a coaxial cylindrical two-zone porous medium with fractal structure is posed and numerically solved. The porous medium is studied in the form of cylinder with two parts: macropore—with high permeability characteristics in the central part and micropore—with low permeability around it. Anomalous solute transport is modeled by differential equations with a fractional derivative. The solute concentration and pressure fields are determined. Based on numerical results, the influence of the fractional derivatives order on the solute transport process is analysed. It was shown that with a decrease in the order of the derivatives in the diffusion term of the transport equation in the macropore leads to a “fast diffusion” in both zones. Characteristics of the solute transport in both zones mainly depend on the concentration distribution and other hydrodynamic parameters in the macropore.

The various mode of stresses for the interaction between two inclined cracks in the upper part of bonded two half planes which are normal stress (Mode I), shear stress (Mode II), tearing stress (Mode III) and mixed stress was studied. For this problem, the modified complex potentials (MCPs) method was used to develop the new system of hypersingular integral equations (HSIEs) by applying the conditions for continuity of resultant force and displacement functions with the unknown variable of crack opening displacement (COD) function and the right hand terms are the tractions along the crack. The curve length coordinate method and Gauss quadrature rules were used to solve numerically the obtained HSIEs to compute the stress intensity factors (SIFs) in order to determine the strength of the materials containing cracks. Numerical solutions presented the characteristic of nondimensional SIFs at the cracks tips. It is obtained that the various stresses and the elastic constants ratio are influences to the value of nondimensional SIFs at the crack tips purposes.

The inclined crack problem in bonded dissimilar materials was considered in this study. The system of hypersingular integral equations (HSIEs) was formulated using the modified complex potentials (MCP) function method, where the continuity conditions of the resultant force and the displacement are applied. In the equations, the crack opening displacement (COD) serves as the unknown function and the traction along the cracks as the right-hand terms. By applying the curved length coordinate method and the appropriate quadrature formulas, the HSIEs are reduced to the system of linear equations. It was found that the nondimensional stress intensity factors (SIF) at the crack tips depend on the ratio of elastic constants, the crack geometries and the distance between the crack and the boundary.

In this paper, numerical solutions of multiple cracks problems in an infinite plate are studied. Hypersingular integral equations (hieq) for the cracks are formulated using the complex potential method. For all kernels such as regular or hypersingular kernels, we are using the appropriate quadrature formulas to solve and evaluate the unknown functions numerically . Furthermore, by using this equation the stress intensity factor (SIF) was calculated for crack tips. For two serial cracks (horizontal) and two dissimilar cracks (horizontal and inclined), our numerical results agree with the previous works.

We present a new third order convergence iterative method for m multiple roots of nonlinear equation. The proposed method requires one evaluation of function and two evaluations of the first derivative of function. In numerical tests exhibit that the present method provides provides high accuracy numerical result as compared to other methods. The stability of the dynamical behaviour of iterative method is investigated by displaying the basin of attraction. Basin of attraction displays less black points which give us wider choices of initial guess in computation.

In this paper, some iterative methods with third order convergence for solving the nonlinear equation were reviewed and analyzed. The purpose is to find the best iteration schemes that have been formulated thus far. Hence, some numerical experiments and basin of attractions were performed and presented graphically. Based on the five test functions it was found that the best method is D87a due Dong’s Family method (Int J Comput Math 21:363–367, 1987)

Using the strain space thermoplasticity theory, proposed by the first author, the coupled dynamic thermomechanical boundary value problems are formulated. The strain space thermoplasticity theory, in contrast to the existing one, allows to formulate the coupled thermoplastic boundary value problems for the displacement and temperature increments. The explicit and implicit finite difference equations for two dimensions case of the boundary value problems are constructed. The numerical solution of the explicit finite difference equations reduced to the application of the recurrent formulas, whereas the implicit scheme reduced to the application of the elimination method. Comparison shows that the numerical results obtained using the explicit and implicit schemes for aforementioned methods are coincides.

Hypersingular integral equation associated with the modified complex potential is formulated to solve the three inclined cracks problems in an elastic half-plane with free traction boundary condition. The modified complex potential possesses two parts; the principal and the complementary parts. The principal part is derived from the original complex potential of the crack problem in an infinite plate. The complementary part eliminates the traction along boundary of half-plane caused by the principal part. The crack opening displacements (COD) is the unknown function and the traction is the right hand terms. The appropriate quadrature formula is adapted to solve the integral equation numerically and the stress intensity factor (SIF)is computed. The behaviour of SIF at crack tips is analysed. Numerical examples show that the SIF increases as the angle of inclined cracks and the distance of cracks from the boundary of half-plane increase. Our results are agreeable with the previous works.

The manual describe and examines modern numerical methods for the numerical solution of partial differential equa¬tions of hyperbolic systems. Hyperbolic equations can be found in many areas of physics and mechanics, such as acoustics, fluid dynamics, elasticity theory, magneto-hydrodynamics, shallow water equations, and others. This book is designed for the students, engineers and researchers who are faced with the necessity of solving hyperbolic systems in various areas of mechanics, physics and applied mathematics. The feature of the manual is to present and classify the different numerical methods expounded on the basis of a single common approach.

We present a new third order convergence iterative method for solving multiple roots of nonlinear equation, which requires one function evaluation and two evaluation of first derivative of function per step. Our present method free from second derivative function. Error term is proved to possess a third order method. Numerical experiments exhibit that our method gives the smallest error of bound per iteration and it is highly accurate as compared to other existing iterative methods.

In this study, a three-point iterative method for solving nonlinear equations is presented. The purpose is to upgrade a fourth order iterative method by adding one Newton step and using a proportional approximation for last derivative. Per iteration this method needs three evaluations of the function and one evaluation of its first derivatives. In addition, the efficiency index of the developed method is √4 8 ≈ 1.682 which supports the Kung-Traub conjecture on the optimal order of convergence. Moreover, numerical and graphical comparison of the proposed method with other existing methods with the same order of convergence are given.

This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y'' (x) = f (x,y, y') including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods.

Two exponentially fitted and trigonometrically fitted explicit two-derivative Runge-Kutta-Nyström (TDRKN) methods are being constructed. Exponentially fitted and trigonometrically fitted TDRKN methods have the favorable feature that they integrate exactly second-order systems whose solutions are linear combinations of functions {exp⁡(wx),exp⁡(-wx)} and {sin⁡(wx),cos⁡(wx)} respectively, when w∈R , the frequency of the problem. The results of numerical experiments showed that the new approaches are more efficient than existing methods in the literature.

Simple and efficient convex homotopy perturbation method (HPM) is presented to obtain an approximate solution of hyper-singular integral equations of the first kind. Convergence and error estimate of HPM are obtained. Three numerical examples were provided to verify the effectiveness of the HPM. Comparisons with reproducing kernel method (Chen et al., 2011) for the same number of iteration is also presented. Numerical examples reveal that the convergence of HPM can still be achieved for some problems even if the condition of convergence of HPM is not satisfied.

In this paper, the problem of arc cracks that lie in the boundary of half circle in an elastic half plane is investigated. The complex potential variables with free traction boundary condition is used to formulate the problem into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function with the help of curve length coordinate method. The numerical results have shown that our results are in good agreement with the previous works. Stress intensity factors for different cracks position are presented .

Modified complex potential with free traction boundary condition is used to formulate the curved crack problem in a half plane elasticity into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function. Numerical examples exhibit the stress intensity factor increases as the cracks getting close to each other, and close to the boundary of the half plane.

In this paper, we derive a new modified third order convergence iterative methods for computing multiple roots of non-linear equations. Our proposed scheme requires one evaluation of function and two evaluations of first derivative. Error term is proved to possess a third order. Numerical performance shows that the proposed method provides a highest accuracy results as compared to other existing third order iterative methods.

In this note, multi-dimensional nonlinear integral equations (multi-dimensional NIEs) in R n is considered. Implementing modified Newton method (Modified NM) reduces multi-dimensional NIEs into multi-dimensional linear integral equations (multi-dimensional LIEs), which can be solved by discritization method. Quadrature method together with collocation are used to find values of unknown functions. Existence and uniqueness solution of the problems are shown. The rate of convergence of the proposed method is proved using the principle of majorant function. Finally, numerical examples are provided and it reveals that proposed method is both accurate and effective as well as comparisons with other methods are also presented.

In this paper, we construct an iterative method with memory based on the Newton–Secants method to solve nonlinear equations. This proposed method has fourth order convergence and costs only three functions evaluation per iteration and without any evaluation of the derivative function. Acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self-accelerator parameter is estimated using Newtons interpolation polynomial of third degree. The order of convergence is increased from 4 to 5.23 without any extra function evaluation. This method has the efficiency index equal to 5.2313≈1.7358. We describe the analysis of the proposed method along with numerical experiments including comparison with the existing methods. Finally, the attraction basins of the proposed method are shown and compared with other existing methods.

In this paper, modified homotopy perturbation method (MHPM) is applied to solve the general Fredholm-Volterra integro-differential equations (FV-IDEs) of order with initial conditions. Selective functions and unknown parameters allowed us to obtain two step iterations. It is found that MHPM is a semi-analytical method for FV-IDEs and could avoid complex computations. Numerical examples are given to show the efficiency and reliability of the method. Proof of the convergence of the proposed method is also given.

The formulation for the curved crack in a finite plate is established. The technique is the curved crack in a finite plate is divided into two sub-problems i.e. the curved crack problem in an infinite plate and the finite plate without crack. For the first problem, the curved problem is formulated into Fredholm integral equation, where as for the second problem the complex boundary integral equations based on complex variables are considered. The solution of the coupled boundary integral equations gives the solution on the domain of the boundary.

Modified homotopy perturbation method (HPM) is used to solve the hypersingular integral equations (HSIEs) of the second kind on the interval [-1, 1] with the assumption that the kernel in the form K(x, t)(x-t)-c0 where K(x, t) is a constant on the diagonal of the domain. This method introduced selective functions as Chebyshev polynomials of second kind and unknown parameters that leads to two step iterations and gives exact solution. Example are presented to prove the efficiency and realiability of the method.

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

The multiple cracks problem in an elastic half-plane is formulated into singular integral equation using the modified complex potential with free traction boundary condition. A system of singular integral equations is obtained with the distribution dislocation function as unknown, and the traction applied on the crack faces as the right hand terms. With the help of the curved length coordinate method and Gauss quadrature rule, the resulting system is solved numerically. The stress intensity factor (SIF) can be obtained from the unknown coefficients. Numerical examples exhibit that our results are in good agreement with the previous works, and it is found that the SIF increase as the cracks approaches the boundary of half plane.

In this note, convex Homotopy perturbation method (HPM) is presented for the approximate solution of the linear Fredholm-Volterra integral and integro-differential equation. Convergence and rate of convergence of the HPM are proved for both equations. Five numerical examples are provided to verify the validity and accuracy of the proposed method. Example reveals that HPM is very accurate and simple to implement for integral and integrodifferential equations.

Formulation in terms of hypersingular integral equations for the interaction between straight and curved cracks in plane elasticity is obtained using the complex variable functions method. The curved length coordinate method and a suitable numerical scheme are used to solve such integrals numerically for the unknown function, which are later used to find the stress intensity factor, SIF.

A new algorithm is presented to provide a general solution for a first type Hyper singular Integral Equation (HSIE). The singular integral has been converted into a regular form by cancelling the singularity and then transforming it into a system of algebraic equation based orthogonal poly- nomials. We applied the convergence method presented by (Obaiys, 2013) to conform the numerical solution of the regularized Hadamard equation, which has shown an absolute agreement with the exact solution within small values of n. The proposed method has been examined by various HSIEs where the displacement function satisfies the HÓ§lder-continuous first derivative. Such equations are particularly important inmany physics and engineering problems, such as fracture mechanics, acoustics and elasticity.

We consider an nxn system of nonlinear integral equations of Volterra type (nonlinear VIEs) arising from an economic model. By applying the Newton-Kantorovich method to the nonlinear VIEs we linearize them into linear Volterra type integral equations (linear VIEs). Uniqueness of the solution of the system is shown. An idea has been proposed to find the approximate solution by transforming the system of linear VIEs into a system of linear Fredholm integral equations by using sub-collocation points. Then the backward Newton interpolation formula is used to find the approximate solution at the collocation points. Each iteration is solved by the Nystrom type Gauss-Legendre quadrature formula (QF). It is found that by increasing the number of collocation points of QF with fewer iterations, a high accurate approximate solution can be obtained. Finally, an illustrative example is demonstrated to validate the accuracy of the method.

The present work is devoted to the proof of uniqueness of the solution of the finite elements scheme in the case of variable coefficients. Finite elements method is applied for the numerical solution of the mixed problem for symmetric hyperbolic systems with variable coefficients. Moreover, dissipative boundary conditions and its stability are proved. Finally, numerical example is provided for the two dimensional mixed problem in simply connected region on the regular lattice. Coding is done by DELPHI7.

This paper studies the method for establishing an approximate solution of nonlinear two dimensional Volterra integral equations (NLTD-VIE). The Newton-Kantorovich (NK) suppositions are employed to modify NLTD-VIE to the sequence of linear two dimensional Volterra integral equation (LTD-VIE). The proper-ties of the two dimensional Gauss-Legenre (GL) quadrature fromula are used to abridge the sequence of LTD-VIE to the solution of the linear algebraic system. The existence and uniqueness of the approximate solution is demonstrated, and an illustrative example is provided to show the precision and authenticity of the method.

The Newton–Kantorovich method (NKM) is widely used to find approximate solutions for nonlinear problems that occur in many fields of applied mathematics. This method linearizes the problems and then attempts to solve the linear problems by generating a sequence of functions. In this study, we have applied NKM to Volterra-type nonlinear integral equations then the method of Nystrom type Gauss–Legendre quadrature formula (QF) was used to find the approximate solution of a linear Fredholm integral equation. New concept of determining the solution based on subcollocation points is proposed. The existence and uniqueness of the approximated method are proven. In addition, the convergence rate is established in Banach space. Finally illustrative examples are provided to validate the accuracy of the presented method.

We develop the Newton-Kantorovich method to solve the system of nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.