scholarly journals Solution of Second-Order IVP and BVP of Matrix Differential Models Using Matrix DTM

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Reza Abazari ◽  
Adem Kılıcman
Keyword(s):  
Boundary Conditions ◽  
Matrix Models ◽  
Boundary Value Problems ◽  
Initial Value Problems ◽  
Present Method ◽  
Initial Conditions ◽  
Transformation Method ◽  
Second Order ◽  
Initial Value ◽  
Matrix Differential

We introduce a matrix form of differential transformation method (DTM) and apply for nonlinear second-order initial value problems (IVPs) and boundary value problems (BVPs) of matrix models which are given by and subject to initial conditions and boundary conditions , where . Also the convergence of present method is established. Several illustrative examples are given to demonstrate the effectiveness of the present method.

scholarly journals SOME ANALOGUES OF A THEOREM OF PEANO FOR BOUNDARY VALUE PROBLEM

1991 ◽  
Vol 22 (1) ◽  
pp. 83-98
Author(s):  
RICK BRANTLEY ◽  
JOHNNY HENDERSON
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Initial Value Problems ◽  
Boundary Value ◽  
Boundary Values ◽  
Initial Value ◽  
Boundary Points

Under certain conditions, solutions of boundary value problems for $y'''=f (x,y, y', y'')$ are differentiated with respect to boundary conditions, both boundary points and boundary values. The results obtained are analogues of one of Peano's theorems on initial value problems.

scholarly journals Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ji Lin ◽  
Yuhui Zhang ◽  
Chein-Shan Liu
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Initial Conditions ◽  
Boundary Value ◽  
Accurate Solution ◽  
Point Boundary ◽  
Third Order ◽  
Boundary Shape ◽  
Free Function ◽  
New Algorithms

AbstractFor nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

scholarly journals Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Jiang Zhu ◽  
Dongmei Liu
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Second Order ◽  
Dynamic Equations ◽  
Maximum Principles ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Approximation Theorems ◽  
Construction Techniques ◽  
Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

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