# Modified Finite Difference Method for solution of two-interval boundary value problems with transition conditions

TL;DR: In this article , a modification of classical Finite Difference Method (FDM) was proposed for the solution of boundary value problems which are defined on two disjoint intervals and involved additional transition conditions at an common end of these intervals.

Abstract: In this study, we have proposed a new modification of classical Finite Difference Method (FDM) for the solution of boundary value problems which are defined on two disjoint intervals and involved additional transition conditions at an common end of these intervals. The proposed modification of FDM differs from the classical FDM in calculating the iterative terms of numerical solutions. To illustrate the efficiency and reliability of the proposed modification of FDM some examples are solved. The obtained results are compared with those obtained by the standart FDM and by the analytical method. Corresponding graphical illustration are also presented.

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TL;DR: In this paper , the authors considered a Sturm-Liouville problem defined on two disjoint intervals together with additional jump conditions across the common endpoint of these intervals, and developed a new tecnique for solving such type nonstandard boundary value problems (BVP).

Abstract: We consider a Sturm-Liouville problem defined on two disjoint intervals together with additional jump conditions across the common endpoint of these intervals. Based on Finite Difference Method (FDM) we have developed a new tecnique for solving such type nonstandard boundary value problems (BVP). To show applicability and effectiveness of the proposed generalization of FDM, we solved a simple but illustrative example. The obtained numerical solutions are graphically compared with the corresponding exact solutions.

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TL;DR: In this paper , the authors considered a Sturm-Liouville problem defined on two disjoint intervals with additional jump conditions across the common endpoint of these two intervals, and developed a new Finite Difference Method (FDM) based method for solving such type nonstandard boundary value problems.

Abstract: We consider a Sturm-Liouville problem defined on two disjoint intervals
together with additional jump conditions across the common endpoint of these
intervals. Based on Finite Difference Method (FDM) we have developed a new
tecnique for solving such type nonstandard boundary value problems (BVP). To
show applicability and effectiveness of the proposed generalization of FDM,
we solved a simple but illustrative example. The obtained numerical
solutions are graphically compared with the corresponding exact solutions.

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06 Sep 2007

TL;DR: This book discusses infinite difference approximations, Iterative methods for sparse linear systems, and zero-stability and convergence for initial value problems for ordinary differential equations.

Abstract: Finite difference approximations -- Steady states and boundary value problems -- Elliptic equations -- Iterative methods for sparse linear systems -- The initial value problem for ordinary differential equations -- Zero-stability and convergence for initial value problems -- Absolute stability for ordinary differential equations -- Stiff ordinary differential equations -- Diffusion equations and parabolic problems -- Addiction equations and hyperbolic systems -- Mixed equations -- Appendixes: A. Measuring errors -- B. Polynomial interpolation and orthogonal polynomials -- C. Eigenvalues and inner-product norms -- D. Matrix powers and exponentials -- E. Partial differential equations.

1,349 citations

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01 Jan 1985

TL;DR: This book discusses Consistency, Stability, and Convergence higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods and Initial Value Methods.

Abstract: List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index.

1,154 citations

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14 Jan 1991TL;DR: This work treats numerical analysis from a mathematical point of view, demonstrating that the many computational algorithms and intriguing questions of computer science arise from theorems and proofs.

Abstract: Taking the time to develop the appropriate theory so readers appreciate the mathematics behind the algorithms, the text has more content but a less formal writing style. The authors' presentation of approximating functions and numerical solution of differential equations are thorough with coverage of splines and boundary value problems. Algorithms are developed in pseudocode (not FORTRAN or Pascal).

982 citations

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TL;DR: In this paper, a boundary functional problem with transmission conditions for ordinary differential-operator equation in Sobolev spaces with a weight was investigated and an isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel basis of a system of root functions of the problem was obtained.

Abstract: We investigate a boundary-functional problem with transmission conditions for ordinary differential-operator equation in Sobolev spaces with a weight. We prove an isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel basis of a system of root functions of the problem. Obtained results in the article are new even in case of Sobolev spaces without the weight.

77 citations

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TL;DR: In this article, the authors investigated various qualitative properties of eigenvalues and corresponding eigenfunctions of one Sturm-Liouville problem with an interior singular point and introduced a new Hilbert space and integral operator in it such a way that the problem under consideration can be interpreted as a spectral problem of this operator.

Abstract: The aim of this study is to investigate various qualitative properties of eigenvalues and corresponding eigenfunctions of one Sturm-Liouville problem with an interior singular point. We introduce a new Hilbert space and integral operator in it such a way that the problem under consideration can be interpreted as a spectral problem of this operator. By using our own approaches we investigate such properties as uniform convergence of the eigenfunction expansions, the Parseval equality, the Rayleigh-Ritz formula, the minimax principle, and the monotonicity of eigenvalues for the considered boundary value-transmission problem (BVTP).

12 citations