Some developments in general variational inequalities

Hydrodynamic and hydromagnetic stability

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

Numerical Methods for Partial Differential Equations

Dynamic optimization of dissipative PDE systems using nonlinear order reduction

The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets

We suggest an iterative method for solving absolute value equation Ax − |x| = b, where \({A\in R^{n\times n}}\) is symmetric matrix and \({b\in R^{n}}\), coupled with the minimization technique. We also discuss the convergence of the proposed method. Some examples are given to illustrate the implementation and efficiency of the method.

https://www.academicjournals.org/article/article1380724814_Noor et al.pdf

On an iterative method for solving absolute value equations

We use uniform cubic polynomial splines to develop some consistency relations which are then used to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second-order boundary value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite difference, and spline methods.

The use of cubic splines in the numerical solution of a system of second-order boundary value problems

We suggest and analyze some new iterative methods for solving the nonlinear equations using the decomposition technique coupled with the system of equations. We prove that new methods have convergence of fourth order. Several numerical examples are given to illustrate the efficiency and performance of the new methods. Comparison with other similar methods is given.

/pdf/some-new-iterative-methods-for-nonlinear-equations-4i3617lpk6.pdf

Some New Iterative Methods for Nonlinear Equations

Quartic spline method for solving fourth order obstacle boundary value problems

In this paper, we use uniform cubic spline polynomials to derive some new consistency relations These relations are then used to develop a numerical method for computing smooth approximations to the solution and its first, second as well as third derivatives for a second order boundary value problem The present method outperforms other collocations, finite-difference and splines methods of the same order Numerical illustrations are provided to demonstrate the practical use of our method

Eisa A. Al-Said

Papers

On an iterative method for solving absolute value equations

The use of cubic splines in the numerical solution of a system of second-order boundary value problems

Some New Iterative Methods for Nonlinear Equations

Quartic spline method for solving fourth order obstacle boundary value problems

Cubic spline method for solving two-point boundary-value problems