Mirk

mirk(<dy1dx, ...>, <bc1, ...>, nlbc,  x_a, x_b, nmesh)

Input:

<dy1dx, dy2dx, ...> is a group of user defined functions of the form dydx(x, y1, y2, ...) => {some expression of x, y1, y2, ...}, which are enclosed with '<>'. These functions defined the all 1st-order differential equations from or converted from the BVP ODE.

<bc1, bc2, ...> is a group of user defined function of the form bc(y1, y2, ...) => {some expression of y1, y2, ...} which are enclosed with '<>'. These functions define all boundary conditions and they are ordered in a way that the left boundary conditions (at boundary of x = x_a) are written first than all of the right boundary conditions (at boundary x = x_b). Note that for boundary condition like y1(x_a) = a needs to be rewritten as bc1(y1,..) => y1 - a.

nlbc is the number of the boundary conditions implemented on the left boundary (where x = x_a)

x_a is a real number for the left boundary location

x_b is a real number for the right boundary location

nmesh is an positive integer implies the preferred mesh grid points for the solutions

Result:

A row vector gives the mesh points of x for the solutions

A matrix gives the solutions y for the mesh points x whose first column is solution for y1, second row for y2, etc.

mirk(<dy1dx, ...>, <bc1, ...>, nlbc,  x_a, x_b, nmesh, torr)

Input:

<dy1dx, dy2dx, ...> is a group of user defined functions of the form dydx(x, y1, y2, ...) => {some expression of x, y1, y2, ...}, which are enclosed with '<>'. These functions defined the all 1st-order differential equations from or converted from the BVP ODE.

<bc1, bc2, ...> is a group of user defined function of the form bc(y1, y2, ...) => {some expression of y1, y2, ...} which are enclosed with '<>'. These functions define all boundary conditions and they are ordered in a way that the left boundary conditions (at boundary of x = x_a) are written first than all of the right boundary conditions (at boundary x = x_b). Note that for boundary condition like y1(x_a) = a needs to be rewritten as bc1(y1,..) => y1 - a.

nlbc is the number of the boundary conditions implemented on the left boundary (where x = x_a)

x_a is a real number for the left boundary location

x_b is a real number for the right boundary location

nmesh is an positive integer implies the preferred mesh grid points for the solutions

torr is a real number for the calculation tolerance

Result:

A row vector gives the mesh points of x for the solutions

A matrix gives the solutions y for the mesh points x whose first column is solution for y1, second row for y2, etc.

mirk(<dy1dx, ...>, <bc1, ...>, nlbc, x)

Input:

<dy1dx, dy2dx, ...> is a group of user defined functions of the form dydx(x, y1, y2, ...) => {some expression of x, y1, y2, ...}, which are enclosed with '<>'. These functions defined the all 1st-order differential equations from or converted from the BVP ODE.

<bc1, bc2, ...> is a group of user defined function of the form bc(y1, y2, ...) => {some expression of y1, y2, ...} which are enclosed with '<>'. These functions define all boundary conditions and they are ordered in a way that the left boundary conditions (at boundary of x = x_a) are written first than all of the right boundary conditions (at boundary x = x_b). Note that for boundary condition like y1(x_a) = a needs to be rewritten as bc1(y1,..) => y1 - a.

nlbc is the number of the boundary conditions implemented on the left boundary (where x = x_a)

x is a row/column vector which gives the preferred mesh points for the solutions. Note that the first and the last element implies the left and right boundaries.

Result:

A row vector gives the mesh points of x for the solutions

A matrix gives the solutions y for the mesh points x whose first column is solution for y1, second row for y2, etc.

mirk(<dy1dx, ...>, <bc1, ...>, nlbc, x, torr)

Input:

<dy1dx, dy2dx, ...> is a group of user defined functions of the form dydx(x, y1, y2, ...) => {some expression of x, y1, y2, ...}, which are enclosed with '<>'. These functions defined the all 1st-order differential equations from or converted from the BVP ODE.

<bc1, bc2, ...> is a group of user defined function of the form bc(y1, y2, ...) => {some expression of y1, y2, ...} which are enclosed with '<>'. These functions define all boundary conditions and they are ordered in a way that the left boundary conditions (at boundary of x = x_a) are written first than all of the right boundary conditions (at boundary x = x_b). Note that for boundary condition like y1(x_a) = a needs to be rewritten as bc1(y1,..) => y1 - a.

nlbc is the number of the boundary conditions implemented on the left boundary (where x = x_a)

x is a row/column vector which gives the preferred mesh points for the solutions. Note that the first and the last element implies the left and right boundaries.

torr is a real number for the calculation tolerance

Result:

A row vector gives the mesh points of x for the solutions

A matrix gives the solutions y for the mesh points x whose first column is solution for y1, second row for y2, etc.

Input

Output

dy1dx(x, y1, y2) => y2

dy2dx(x, y1, y2) => 5 * sinh(5 * y1)

bc1(y1, y2) => y1

bc2(y1, y2) => y1 - 1

mirk(<dy1dx, dy2dx>, <bc1, bc2>, 1, 0, 1, 20)