A Discreate Calculus with Applications of High-Order Discretizations to Boundary-Value Problems

2004 ◽  
Vol 4 (2) ◽  
pp. 228-261 ◽  
Author(s):  
Stanly Steinberg
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Energy Inequality ◽  
Numerical Test ◽  
Mixed Boundary Conditions ◽  
Discrete Analog ◽  
High Order ◽  
Sturm Liouville

AbstractWe develop a discrete analog of the differential calculus and use this to develop arbitrarily high-order approximations to Sturm–Liouville boundary-value problems with general mixed boundary conditions. An important feature of the method is that we obtain a discrete exact analog of the energy inequality for the continuum boundary-value problem. As a consequence, the discrete and continuum problems have exactly the same solvability conditions. We call such discretizations mimetic. Numerical test confirm the accuracy of the discretization. We prove the solvability and convergence for the discrete boundary-value problem modulo the invertibility of a matrix that appears in the discretization being positive definite. Numerical experiments indicate that the spectrum of this matrix is real, greater than one, and bounded above by a number smaller than three.

scholarly journals Numerical Results for Linear Sequential Caputo Fractional Boundary Value Problems

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 910
Author(s):  
Bhuvaneswari Sambandham ◽  
Aghalaya S. Vatsala ◽  
Vinodh K. Chellamuthu
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Conditions ◽  
Numerical Code ◽  
Fractional Boundary Value Problem ◽  
Monotone Method ◽  
Fractional Boundary Value Problems

The generalized monotone iterative technique for sequential 2 q order Caputo fractional boundary value problems, which is sequential of order q, with mixed boundary conditions have been developed in our earlier paper. We used Green’s function representation form to obtain the linear iterates as well as the existence of the solution of the nonlinear problem. In this work, the numerical simulations for a linear nonhomogeneous sequential Caputo fractional boundary value problem for a few specific nonhomogeneous terms with mixed boundary conditions have been developed. This in turn will be used as a tool to develop the accurate numerical code for the linear nonhomogeneous sequential Caputo fractional boundary value problem for any nonhomogeneous terms with mixed boundary conditions. This numerical result will be essential to solving a nonlinear sequential boundary value problem, which arises from applications of the generalized monotone method.

scholarly journals Hartman-Wintner and Lyapunov-type inequalities for high order fractional boundary value problems

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2273-2281
Author(s):  
Şuayip Toprakseven
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Nontrivial Solution ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
High Order ◽  
Fractional Boundary Value Problem ◽  
Sturm Liouville ◽  
Fractional Boundary Value Problems

In this paper, we obtain Hartman-Wintner and Lyapunov-type inequalities for the three-point fractional boundary value problem of the fractional Liouville-Caputo differential equation of order ? 2 (2; 3]. The results presented in this work are sharper than the existing results in the literature. As an application of the results, the fractional Sturm-Liouville eigenvalue problems have also been presented. Moreover, we examine the nonexistence of the nontrivial solution of the fractional boundary value problem.

scholarly journals The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

2020 ◽  
Vol 12 (1) ◽  
pp. 173-188
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Boundary Conditions ◽  
Multipoint Problem ◽  
Uniqueness Of The Solution

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

Positive and maximal positive solutions of singular mixed boundary value problem

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Ravi Agarwal ◽  
Donal O’Regan ◽  
Svatoslav Staněk
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Existence Results ◽  
Mixed Boundary Value Problem ◽  
Time Singularity ◽  
Strong Singularity

AbstractThe paper is concerned with existence results for positive solutions and maximal positive solutions of singular mixed boundary value problems. Nonlinearities h(t;x;y) in differential equations admit a time singularity at t=0 and/or at t=T and a strong singularity at x=0.

scholarly journals Multiplicity of positive solutions for second order Sturm-Liouville boundary value problems

2016 ◽  
Vol 25 (2) ◽  
pp. 215-222
Author(s):  
K. R. PRASAD ◽  
◽  
N. SREEDHAR ◽  
L. T. WESEN ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Multiple Positive Solutions ◽  
Sturm Liouville ◽  
Multiplicity Of Positive Solutions

In this paper, we develop criteria for the existence of multiple positive solutions for second order Sturm-Liouville boundary value problem, u 00 + k 2u + f(t, u) = 0, 0 ≤ t ≤ 1, au(0) − bu0 (0) = 0 and cu(1) + du0 (1) = 0, where k ∈ 0, π 2 is a constant, by an application of Avery–Henderson fixed point theorem.

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