On a first order partial differential equation with the nonlocal boundary condition

Abstract

In this paper, the initial value problem for the first order partial differential equation with the nonlocal boundary condition is studied. In applications, the stability estimates for the first order partial differential equation with the nonlocal boundary condition are obtained. The finite difference method for the initial value problem for hyperbolic equations with nonlocal boundary conditions is applied. In practice, the stability estimates for the solution of the difference scheme of the problem for hyperbolic equations with nonlocal boundary conditions are obtained.

A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

“…Traces of ordinary differential equations can be found in various fields of mathematics, natural, or social sciences. Geometry, various engineering fields including analytical mechanics and electrical engineering, geology, physics, chemistry (in the analysis of nuclear chain reactions), biology (in the modeling of infectious diseases and genetic changes), ecology (in population modeling), and economy (in the modeling of dividend and stock price changes) are some of the scientific branches in which the ordinary differential equations play an essential role [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Since most of the differential equations that provide a relatively accurate model of the target phenomena have a complex and nonlinear form, finding an analytical solution for these problems is usually very difficult or even impossible.…”

“…− 348,000v 3 + 94,500v cos(v) 7 + −441,600v 4 + 207,000v 2 −27,000) sin(v) − 1008v 9 − 2592v 7 + 810v 5 cos(v) 6 + 216v 10 − 42v 8 − 2430v 6 sin(v) +360v 9 + 1326v 7 − 71,030v 5 + 546,000v 3 − 168,750v cos(v) 5 + 960v 8 + 2700v 6 −66,750v 3 cos(v) + −84v 8 − 252v 6 − 60,000v 4 + 13,500v 2 sin(v) − 324v 9 + 192v7 …”

“…− 348,000v3 + 94,500v cos(v)7 + −441,600v 4 + 207,000v 2 −27,000) sin(v) − 810v 9 − 2790v 7 + 810v 5 cos(v)6 + 324v 10 − 150v 8 − 2430v 6 sin(v) +360v 9 + 1326v 7 − 71,030v 5 + 546,000v 3 − 168,750v cos(v) 5 + 960v 8 + 2700v 6 +422,400v 4 − 243,000v 2 + 40,500 sin(v) − 684v 9 + 7221v 7 − 1143v 5 cos(v) 4 + 162v 10 −3309v 8 + 3303v 6 sin(v) + 360v 9 − 3966v 7 − 256,718v 5 − 124,500v 3 + 74,250v cos(v) 3 + 1824v 8 − 3204v 6 + 115,200v 4 + 22,500v 2 − 13,500 sin(v) + 2313v 9 − 5064v 7 +369v 5 cos(v) 2 + −486v 10 + 1569v 8 + 153v 6 sin(v) − 1530v 9 + 3018v 7 + 125,348v 5 −66,750v 3 cos(v) + −84v 8 − 252v 6 − 60,000v 4 + 13,500v 2 sin(v) − 252v 9 + 120v 7…”

An efficient four‐step multiderivative method for the numerical solution of second‐order IVPs with oscillating solutions

“…The new method is from the second class. The purpose of this article is to introduce more efficient methods for the numerical solution of second‐order initial value problems with highly oscillatory solutions (see, eg, References 5‐36,37). To be more specific, the main aim of this article is to develop a new four‐step predictor P‐stable method which contains phase‐lag and some of the derivatives therein are equal to zero.…”

Positivity of a differential operator with nonlocal conditions

“…Various local and nonlocal boundary value problems for partial differential equations can be considered as an abstract boundary value problem for ordinary differential equations in a Banach space with a densely defined unbounded space operator. The theory of differential and difference operators and their related applications has been investigated by many researchers (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]).…”