Partial Differential Equations (PDEs)

Mathematical Science is filled with equations and one of the most important types of equations is Partial Differential Equations (PDEs). It is an equation in which we find various unknown variables and their partial derivatives are also hidden in the same equation. These equations are used to understand the change in time and space of a quantity and are helpful in different fields like weather forecasting, fluid dynamics, etc. In this article, we will understand in detail PDEs, their examples, the Classical Cauchy problem, and other related aspects.

Meaning and Examples of Partial Differential Equations

Partial differential equations are defined as those equations which comprise certain unknown functions and their derivatives. The function is usually made up of at least two variables and is in the form of u(x,y). In the context of PDE, a Partial solution is that solution that will satisfy a given PDE. 

An example of partial differential equation for a function u(t, x, y) can be: 

∂u/∂t = D=(∂2u/∂x2 + ∂2u/∂y2). 

Here u is the function and t, x, and y are multiple variables in one equation and their derivative is also in the same equation. 

Types of Partial Differential Equations

Partial differential equations can be classified into the following types depending on the number of variables and solutions of the differential equation. 

• Division on basis of the shape of function and relation of a,b and c 

1. Elliptic Partial differential equations: An equation is elliptic if the relation between the constants of the function is in the form of (square of b- ac less than zero). 
2. Parabolic Partial differential equations: For an equation in becoming a parabolic partial equation, it needs to satisfy the condition b2-ac=0. 
3. Hyperbolic PDE differential equations: If an equation is hyperbolic then b2-ac>0. 

• Division on basis of the order of derivative 

1. First-Order Partial Differential Equations: If the partial differential equation contains only the first derivative of a function, it is called First-Order Partial Differential Equations. An example of this type of equation is F(x1, x2…xn, u) equals zero.
2. Linear Partial Differential Equation: In this kind of differential equation, all the derivatives and the variables occur in a linear form. Just opposite to this kind of equation is a non-linear partial differential equation that does not contain the derivatives and variables in a linear form. 
3. Quasi-linear Differential equations: Quasi means resembling something. Therefore a quasi-linear equation resembles a linear equation. In this kind of equation, the derivatives occur in a linear form and the coefficients occurring with the highest order of the equation will occur in a particular order.
4. Homogeneous equations: An equation is said to be a homogeneous partial differential equation if all terms have similar features. For example, if all the terms of the equation are attached with a partial derivative or something else which creates a similarity in all the terms, it is called a homogeneous equation.
5. Second-Order Partial Differential Equations: Those equations in which the highest order of the function belongs to the second order. The function in this equation is given as A(x,y)uxx +b (x,y)uxy+…. G(x,y)u= f(x,y) are called Second-Order Partial Differential Equations. Here a, b, c…g will represent the coefficients and with each alphabet, the function of u will change. This means that with 

• c coefficient, yx will be used in place of xy, 
• with d, yy will be used 
• With e, ux will be written 
• With f, it will be uy

Use of Partial Differential Equations (PDE): 

Partial differential equations are often used to make models of multidimensional systems. This is why these equations are used in the explanation of several phenomena like electrodynamics, electrostatics, heat, sound, and much more. Some of the most commonly applied partial differential equations in different fields are:

1. Partial differential Equation for heat conduction: ∂T/ ∂t = C [∂2T/ ∂x2].
2. Partial differential equation for single dimensional wave: ∂2u/ ∂t2 = C2[∂2u/∂x2]. 

Classical Cauchy problem in PDE

In the case of PDE, the classical Cauchy problem is the one where certain conditions of the hypersurface are fulfilled. According to the Classical Cauchy problem theorem, a unique solution will lie for all analytical functions. To solve Partial differential equations, the following steps are used: 

1. Both the sides are differentiated w.r.t. X and y. 
2. You will get two equations which will be again differentiated for a different variable. 

Iii. Both the equations are multiplied with x and y respectively. In the end, you will get the answer. 

How are ordinary differential equations different from partial differential equations? 

Ordinary differential equations are based on single dimension systems but partial differential equations use multi-dimensional systems. The variable count in both these equations is also different. Partial differential equations contain multiple variables but ordinary differential equations usually have one variable whose derivative is also present in the same equation. 

Conclusion

Partial differential equations play a major role in solving several mathematical and scientific problems and phenomena. These equations are used to create a relationship between a function and its derivative which are present in the same equation. With the right steps given above, you can solve any PDE.