First order non-linear partial differential equation & its applications
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There are five types of methods for solving first order non-linear partial differential equations: I) Equations containing only p and q variables. II) Equations relating z as a function of u. III) Equations that can be separated into functions of single variables. IV) Clairaut's Form where the solution is directly substituted. V) Charpit's Method which is a general method taking integrals of auxiliary equations to solve dz=pdx+qdy and find the solution. These types cover a range of applications including Poisson's, Helmholtz's, and Schrödinger's equations in fields like electrostatics, elasticity, wave theory and quantum mechanics.
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We reflect on whether JRR Tolkien’s Lord of the Rings Influenced by Nordic Mythology, Homer’s Iliad, and/or Catholicism:
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First order non-linear partial differential equation & its applications
- 1. There are five ways of non-linear partial differential equations of first order and their method of solution as given below. •Type I: •Type II: •Type III: (variable separable method) •Type IV: Clairaut’s Form •CHARPIT’S METHOD
- 2. Type I: f(p, q)=0 Equations of the type f(p, q)=0 i.e. equations containing p and q only Let the required solution be and Substituting these values in , f(p, q)=0 we get f(a, b)=0 From this, we can obtain b in terms of a (or) a in terms of b Let , then the required solution is
- 3. Type II: Let us consider the Equations of the type Let z is a function of u ie. z = and Now, is the 1st order differential equation in terms of dependent variable z and independent variable u. Solving this differential equation and finally substitute gives the required solution. u z a y u u z y z q ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ =
- 4. Type III – f1(p,x) = f2(q,y) (variable separable method) Let us consider differential equation of the form f1(p,x) = f2(y,q) Let f1(p,x) = f2(y,q) = a (say) Now f1(p,x) = a → p = Ψ1(x) (ie. Writing p in terms of x) f2(q,y) = a→ q = Ψ2(y) (ie. Writing q in terms of y) Now, dz = = pdx + qdy so dz = Ψ1(x)dx + Ψ2(y)dy dy y z dx x z ∂ ∂ + ∂ ∂
- 5. Type IV: Clairaut’s Form Equations of the form Let the required solution be then Required solution is i.e. Directly substitute a in place of p and b in place of q in the given equation.
- 6. CHARPIT’S METHOD This is a general method to find the complete integral of the non- linear PDE of the form f (x , y, z, p, q) = 0 Now Auxillary Equations are given by Here we have to take the terms whose integrals are easily calculated, so that it may be easier to solve and finally substitute in the equation dz = pdx + qdy Integrate it, we get the required solution. Note :that all the above (TYPES) problems can be solved in this method.
- 7. Applications of PDE • Poisson’s Equation which arises in electrostatics, elasticity theory and elsewhere. • Helmholtz's Equation which arisis in wave theory. • Schrödinger's Equation which arises in quantum mechanics.
- 8. THANK YOU