Partial differential equations are used to mathematically formulate, andthus aid the solution of, physical and other problems involving functionsof several variables, such as the propagation of heat or sound, fluid flow,elasticity, electrostatics, electrodynamics, etc.
A first-order quasilinear partial differential equation with two independent variables has the general form\[\tag1f(x,y,w)\frac\partial w\partial x+g(x,y,w)\frac\partial w\partial y=h(x,y,w).\]
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Example. Consider the linear equation\[\frac\partial w\partial x+a\frac\partial w\partial y=b.\]The associated characteristic system of ordinary differential equations\[\fracdx1=\fracdya=\fracdwb\]has two integrals\[y-ax=C_1,\quad \ w-bx=C_2.\]Therefore, the general solution to this PDE can be written as \(w-bx=\Psi(y-ax)\), or\[w=bx+\Psi(y-ax),\]where \(\Psi(z)\) is an arbitrary function.
A second-order linear partial differential equation with two independent variables has the form\[\tag9a(x,y)\frac\partial^2w\partial x^2+2b(x,y)\frac\partial^2w\partial x\,\partial y+c(x,y)\frac\partial^2w\partial y^2=\alpha(x,y)\frac\partial w\partial x+\beta(x,y)\frac\partial w\partial y+\gamma(x, y)w+\delta(x,y).\]
A second-order semilinear partial differential equation with two independent variables has the form\[\tag10a(x,y)\frac\partial^2w\partial x^2+2b(x,y)\frac\partial^2w\partial x\,\partial y+c(x, y)\frac\partial^2w\partial y^2=F\biggl(x,y,w,\frac\partial w\partial x,\frac\partial w\partial x\biggr).\]
In the general case, a second-order nonlinear partial differentialequation with two independent variables has the form\[F\biggl(x,y,w,\frac\partial w\partial x,\frac\partial w\partial y,\frac\partial^2w\partial x^2,\frac\partial^2w\partial x\,\partial y,\frac\partial^2w\partial y^2\biggr)=0.\]
The classification and the procedure for reducing linear and semilinearequations of the form (9) and (10) to a canonical form are only determined by the left-hand side of the equations (see below for details).
Any semilinear partial differential equation of the second-order with twoindependent variables (10) can be reduced, by appropriate manipulations, toa simpler equation that has one of the three highest derivativecombinations specified above in examples (11), (12), and (14).
Most PDEs of mathematical physics govern infinitely many qualitativelysimilar phenomena or processes. This follows from the fact thatdifferential equations have, as a rule, infinitely many particular solutions.The specific solution that describes the physical phenomenon under study isseparated from the set of particular solutions of the given differentialequation by means of the initial and boundary conditions.
Substituting the values set on the characteristics (25) into the generalsolution of the wave equation (13), one arrives at a system of linearalgebraic equations for \(\varphi(x)\) and \(\psi(x)\ .\) As a result, thesolution to the Goursat problem (12), (25) is obtained in the form\[w(x,t)=\varphi\biggl(\fracx+t2\biggr)+\psi\biggl(\fracx-t2\biggr)-\varphi(0).\]The solution propagation domain is the parallelogram bounded by the fourlines\[x-t=0,\quad x+t=0,\quad x-t=2b,\quad x+t=2a.\]
In general, the nonlinear heat equation (27) admits exact solutions of the form\[\beginarrayllw=W(kx-\lambda t)& (\hboxtraveling-wave solution),\\w=U(x/\!\sqrt t\,)& (\hboxself-similar solution),\endarray\]where \(W=W(z)\) and \(U=U(r)\) are determined by ordinary differential equations, and\(k\) and \(\lambda\) are arbitrary constants.
Equations of this form arise in differential geometry and various areas ofphysics (superconductivity, dislocations in crystals, waves inferromagnetic materials, laser pulses in two-phase media, and others). For\(f(w)\equiv 0\) and \(a=1\), equation (31) coincides with the linear waveequation (12).
1. In general, the nonlinear heat equation (32)admits exact solutions of the form\[\beginarrayllw=W(z),& z=k_1x+k_2y,\\w=U(r),& r=\sqrt(x+C_1)^2+(y+C_2)^2,\endarray\]where \(W=W(z)\) and \(U=U(r)\) are determined by ordinary differential equations, while\(k_1\), \(k_2\), \(C_1\), and \(C_2\) are arbitrary constants.
In order that equation (39) coincides with (38), one must require thatthe powers of \(C\) are the same, which yieldsthe following system of linear algebraic equations for the constants \(k\)and \(m\ :\)\[m-1=m-2k=mn.\]This system admits a unique solution\[\,k=\frac 12\ ,\]\(m=\frac 11-n\ .\) Using this solution together with relations (35)and (37), one obtains self-similar variables in the form\[w=t^1/(1-n)U(\zeta),\quad \ \zeta=xt^-1/2.\]Inserting these into (38), one arrives at the following ordinary differentialequation for \(U(\zeta)\ :\)\[aU''_\zeta\zeta+\frac12\zeta U'_\zeta+\frac 1n-1U+bU^n=0.\]
Equations (49), (51), (53) and (55) constitute the full system of equations for thecalculation of the numerical solution to Eq. (41). Note that we have replacedthe original PDE, Eq. (41), with a set of approximating algebraic equations(Eqs. (49), (51), (53) and (55)) which can easily be programmed for a computer. 2ff7e9595c
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