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detools::pdesolve -- solver for partial differential equations

Introduction

detools::pdesolve solves partial differential equations.

Call(s)

detools::pdesolve(pde, indl, depl)
detools::pdesolve(df, DV)

Parameters

pde - the differential equation(s): either a single expression or a list of expressions.
indl - the independent variables: a list of (indexed) identifiers.
depl - the dependent variables: a list of (indexed) identifiers.
df - the differential equation(s): either a single element of a domain in Cat::DifferentialFunction(DV) or a list of such elements.
DV - the differential variables: a domain in Cat::DifferentialVariable.

Returns

a single expression or a list of expressions; each entry represents a component of the solution. If detools::pdesolve is not able to solve the equation, it is returned unchanged.

Related Functions

detools::charSolve, detools::detSys, solve

Details

Example 1

The following call solves the quasi-linear equation diff(u,t)-u*(diff(u,x)+diff(u,y))=0 with the initial condition u(t=0)=x+y. For the input of the differential equation a condensed notation is used (u([t]) instead of diff(u(x,y,t),t) etc.); the initial condition is given in parametrized form.

>> detools::pdesolve(u([t]) - u*(u([x]) + u([y])), [t, x, y], [u], 
            {t = 0, x = sigma, y = tau, u = sigma + tau}, 
            {sigma, tau})
                                   2 (t x - x - t y)
                       u = y - x + -----------------
                                        2 t - 1

One can easily check by entering this expression into the differential equations that it is indeed a solution (don't forget to use normal!) and that for t=0 we have u=x+y as required by the initial condition.




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