PERIDYNAMICS_1D_STEADY a MATLAB library which solves a 1D steady version of the Poisson equation, using the non-local peridynamics model, by Marta D'Elia.

The problem data is specified by a user-supplied file which evaluates:

• the exact solution u(x);
• the derivative of the exact solution u'(x).
• the right hand side of the Poisson equation, -u"(x).
• the lifting, a function which adjusts the problem for nonzero Dirichlet boundary conditions;

### Languages:

PERIDYNAMICS_1D_STEADY is available in a MATLAB version.

### Related Data and Programs:

PERI1D, a C program which sets up and solves a 1D time-dependent peridynamics problem, by Miroslav Stoyanov;

Marta D'Elia

### Reference:

1. Qiang Du, Max Gunzburger, Rich Lehoucq, Kun Zhou,
Analysis and approximation of nonlocal diffusion problems with volume constraints,
SIAM Review,
Volume 54, Number 4, pages 667-696, 2012.
2. Max Gunzburger, Rich Lehoucq,
A nonlocal vector calculus with application to nonlocal boundary value problems,
Multiscale Modeling and Simulation,
Volume 8, Number 5, 2010, pages 1581-1598.

### Examples and Tests:

• pd1d_steady_test.m, calls both convergence tests for all problems. You probably don't want to call this function, but it shows you how to call either test below for any of the five sample problems.
• pd1d_steady_test01.m, for a given problem (1, 2, 3, 4 or 5), this test computes convergence results as EPSILON and H go to zero in fixed ratio. You specify the problem as an input quantity that is a MATLAB function handle, such as "@problem1".
• pd1d_steady_test02.m, for a given problem (1, 2, 3, 4 or 5), this test computes convergence results for fixed EPSILON, while H goes to zero. You specify the problem as an input quantity that is a MATLAB function handle, such as "@problem5".

PROBLEM 1 has the solution U(X) = X^2, on the domain [0,1].

PROBLEM 2 has the solution U(X) = X^2*(1-X^2), on the domain [0,1].

PROBLEM 3 has the solution U(X) = X^2, on the domain [0,1].

PROBLEM 4 has the solution U(X)=X-1/4 left of 1/2, and U(X)=X-1/2 to the right of 1/2, on the domain [0,1]. This problem was devised to study the behavior of singularities.

PROBLEM 5 has the solution U(X) = 1+X, on the domain [0,1]. This problem was devised for a simple accuracy check.

You can go up one level to the MATLAB source codes.