Unit - 1

Partial Differential Equations

1.1 Partial Differential Equations - Basic concepts and Definitions & Mathematical Problems

Introduction

An equation containing the derivatives or differentials of one or more dependent variables with respect to one or more independent variables is called a differential equation.

Example:

Notation:

………………..

Order of a differential equation:

The order of the highest derivative involved in the differential equation i.e. how many times it is differentiated is called the order of the differential equation.

Degree of the differential equation:

The degree of the highest derivative involved in the differential equation after made free from radicals and fractions as far as derivatives are concerned i.e.  power of the highest derivative is called  the degree of the differential equation.

Example: has order 2 and degree 3.

Types of Differential equations:

Ordinary Differential Equation:

A differential equation involving derivatives with respect to single independent variables is called ordinary differential equation.

Example:

Partial Differential Equation:

A differential equation involving derivatives with respect to more than one independent variable is called partial differential equation.

Example:

Linear and Non-linear differential equations:

A differential equation is linear if

1. Every dependent variable and every derivatives present in the equation is only of degree one.
2. No products of dependent variables or derivatives occur in the equation.

Otherwise it is a non-linear equation.

Solution of differential equations:

A relation between the dependent variable and independent variables which satisfies the given differential equation is called the solution or the integral of the differential equation.

Example: The relation     is the solution of .

General solution, Particular   solution and Singular solution:

Let ….(i)

Be an nth order ordinary differential equation.

General Solution: A solution of (i) which contain n arbitrary constants is called general solution.

Particular Solution: A solution of (i) which is obtained by substituting particular values to the one or more n arbitrary constants is called the particular solution.

Singular solution: A solution which cannot be obtained from the general solution.

Formation of Differential equation:

Working Rule: To construct the differential equation from the given family of curve in x and y containing n arbitrary constants.

1. Write the given family of curves.
2. Differentiate n times to get n equations.
3. Eliminate the n arbitrary constant using (a) and (b).

Example1: Find the differential equation of all circles touching the y-axis at the origin?

The equation of the circle that touches y-axis at the origin is

….(i)

Differentiating (i) with respect to x, we get

Or

Or

Or

Or 

Or

This is the required differential equation.

Example2: Form the differential equation of ?

Given curve

Differentiating the above curve with respect to x.

Multiplying both side by x.

Now, 

=0

Hence the required equation is

Partial differential equations-

A partial differential equation (PDE) is an equation involving one or more partial derivatives of an (unknown) function, call it u, that depends on two or more variables, often time t and one or several variables in space. The order of the highest derivative is called the order of the PDE. Just as was the case for ODEs, second-order PDEs will be the most important ones in applications.

Just as for ordinary differential equations (ODEs) we say that a PDE is linear if it is of the first degree in the unknown function u and its partial derivatives. Otherwise we call it nonlinear.

The standard methods of solving the differential equations of the following

Types:

(i)                Equations solvable by separation of the variables.

(ii)              Homogeneous equations.

(iii)           Linear equations of the first order.

(iv)            Exact differential equations.

Definition- A partial differential equation is one in which y depends on two or more independent variables say x, t , . . . (so that the derivatives of y are partial derivatives.)

The independent variables will be denoted by x and y and the dependent variable by z. The partial differential coefficients are denoted as-

The general solution of non-homogeneous second order linear P.D.E. With constant coefficients is obtained as the sum of complementary function and particular integral.

A partial differential equation is an equation involving two (or more) independent variables x, y and a dependent variable z and its partial derivatives like- .

Order of a partial differential equation (P.D.E.) is the order of the highest ordered derivative appearing in the partial differential equation.

A PDE is said to be quasi-linear if the derivatives of the highest order which occur in the equation are linear.

A quasi-linear PDE is called semi-linear if the coefficients of the highest order derivatives do not contain either the dependent variable or its derivative.

A semi-linear PDE is called linear if it is linear in the dependent variable and its derivative.

Note- when a PDE is not quasi-linear PDE then it is said to be non-linear.

Order of a differential equation:

The order of the highest derivative involved in the differential equation i.e. how many times it is differentiated is called the order of the differential equation.

Degree of the differential equation:

The degree of the highest derivative involved in the differential equation after made free from radicals and fractions as far as derivatives are concerned i.e.  power of the highest derivative is called  the degree of the differential equation.

Example: has order 2 and degree 3.

Key takeaways

1. The order of the highest derivative involved in the differential equation.
2. The degree of the highest derivative involved in the differential equation after made free from radicals and fractions as far as derivatives are concerned.
3. A differential equation involving derivatives with respect to single independent variables is called ordinary differential equation.
4. A partial differential equation (PDE) is an equation involving one or more partial derivatives of an (unknown) function.
5. A PDE is said to be quasi-linear if the derivatives of the highest order which occur in the equation are linear.
6. A semi-linear PDE is called linear if it is linear in the dependent variable and its derivative.

1.2 First- Order Equations: Classification, Construction and Geometrical Interpretation

Classification of first order partial differential equations-

1. A first order PDE is called linear equation if it is linear in p, q and z.

It is written as-

2.     A first order PDE is called semi-linear equation if it is linear in p and q are functions of x and y only if it is of the form

3.     A first order PDE is called quasi-linear equation if it is linear in p and q. It is of the form

Formation of PDE-

Let

f (x, y, z, a, b) = 0  …… (1)

Be an equation involving two arbitrary constants a and b. Differentiating this equation partially with respect to x and y, we get

By eliminating a, b from (1), (2), (3), we get an equation of the form

F (x, y, z, p, q) = 0 …………..(4)

Which is a partial differential equation of first order.

Note-

1. If the number of arbitrary constants equals to the number of independent variables in (1), then the PDE obtained by elimination is of first order.
2. If the number of arbitrary constants is more than the number of independent variables then the PDE obtained is of second or higher orders.

Method of elimination of arbitrary constants-

Example: Form a partial differential equation from-

Sol.

Here we have-

It contains two arbitrary constants a and c

Differentiate the equation with respect to p, we get-

Or

Now differentiate the equation with respect to q, we get-

Now eliminate ‘c’,

We get

Now put z-c in (1), we get-

Or

Example: Obtain PDE of by eliminating the arbitrary constants

Sol:

Differentiating partially with respect to x and y, we get

Or

Eliminating the two arbitrary constants a and b

Example: Find the differential equation of all spheres whose centres lie on the z-axis.

Sol:

The equation is - , here a and b are the constants.

Now on differentiating, we get

From the second equation

Now put this value in equation first, we get

The second method we use is method of elimination of arbitrary function.

Solution of partial differential equation by direct partial Integration-

Example: Solve-

Sol.

Here we have-

Integrate w.r.t. x, we get-

Integrate w.r.t. x, we get-

Integrate w.r.t. y, we get-

Example: Solve the differential equation-

Given the boundary condition that-

At x = 0,

Sol.

Here we have-

On integrating partially with respect to x, we get-

Here f(y) is an arbitrary constant.

Now form the boundary condition-

When x = 0,

Hence-

On integrating partially w.r.t.x, we get-

Example: Solve the differential equation-

Given that when y = 0, and u = when x = 0.

Sol.

We have-

Integrating partially w.r.t. y, we get-

Now from the boundary conditions,

Then-

From which,

It means,

On integrating partially w.r.t. x givens-

From the boundary conditions, u = when x = 0

From which-

Therefore the solution of the given equation is-

Linear Equations of the First Order

A linear partial differential equation of the first order, commonly known as Lagrange’s Linear equation is of the form

Pp + Qq = R              (1)

Where, P, Q and R are functions of x, y, z. This equation is called a quast linear equation. When P, Q and R are independent of z it is known as linear equation.

Such asn equation is obtained by eliminating an arbitrary function from

Where u,v are are some functions of x, y, z.

Differentiating (2) partially with respect to x and y

Eliminating and , we get

Which simplifies to

This is of the same form as (1)

Now suppose u = a and v=b, where a, b are constants, so that

By cross multiplication we have,

The solution of these equations are u = a and v = b

Therefore, is the required solution of (1).

Thus to solve the equation Pp + Qq =R.

(i) Form the subsidiary equations

(ii) Solve these simultaneous equations

(iii) Write the complete solution as or u=f(v)

Example. Solve   

Solution. Rewriting the given equation as

The subsidiary equations are

The first two fractions give

Integrating we get    n    (i)

Again the first and third fraction give xdx = zdz

Integrating, we get

Hence from (i) and (ii), the complete solution is

Example. Solve

Solution. Here the subsidiary equations are

Using multipliers x,y, and z we get each fraction =

which on integration gives

Again using multipliers l, m and n we get each fraction

which on integration gives lx +my +nz = b     (ii)

Hence from (i) and (ii) the required solution is

Example. Solve

Solution. Here the subsidiary equations are

From the last two fractions, we have

Which on integration gives log y = log z + log a or y/z=a     (i)

Using multipliers x, y and z we have

Each fraction

Which on integration gives

Hence from (i) and (ii) the required solution is

Charpit’s method

This is the general method for finding the complete integral of a non-linear partial differential equation.

Let us consider the equation-

f(x, y, z , p, q) = 0 ………. (1)

Since z depends on x and y, we have-

The main thing in Charpit’s method is to find the another relationship between the variables x, y, z and p. q.

And let the relation be-

Here on solving equation (1) and (2), we get the values of p and q.

When we substitute these values of p and q in (2), it becomes integrable.

To determine , equation (1) and (3) are differentiated with respect to x and y.

We get, when we differentiate with respect to x,

We get, when we differentiate with respect to y,

Eliminating between the equations we get from differentiating for x, we get

Or

…………(4)

Eliminating between the equations we get from differentiating for y, we get

…………(5)

Adding (4) and (5) and keeping in view the relation on, the terms of the last brackets of (4) and (5) cancel. On rearranging, we get-

Or

This equation is the Lagrange’s linear equation of the first order with x, y, z, p, q as independent variables and equation (4) as dependent variable.

Its subsidiary equations are-

An integral of these equations involving p or q or both can be taken as the relation (3) which along with (1) will give the values of p and q to make (2) integrable.

Example: Solve-

Sol.

Let
 

Charpit’s subsidiary equations are-

So that- dq = 0 or q = a

On putting q = a in (1) we get-

Such that-

Integrating

Or

Which is the required solution.

Example: Solve-

Sol.

Let

Charpit’s subsidiary equations are-

From the first and fourth ratios,

Substituting p = a – x in the given equation, we get-

So that-

Multiply both sides by ,

Integrating-

Or

Which is the required solution.

Key takeaways

1. A first order PDE is called linear equation if it is linear in p, q and z.

It is written as-

2.     A first order PDE is called quasi-linear equation if it is linear in p and q. It is of the form

3.     If the number of arbitrary constants is more than the number of independent variables then the PDE obtained is of second or higher orders.

4.     A linear partial differential equation of the first order, commonly known as Lagrange’s Linear equation is of the form

Pp + Qq = R              (1)

Where, P, Q and R are functions of x, y, z.

1.3 Method of Characteristics for obtaining General Solution of Quasi Linear Equations

The general form of a first order partial differential equation is

F (x, y, z, p, q) = 0           ….(1)

Where x, y are the two independent variables, z is the dependent variable and p = and

Complete solution

Any function

f (x, y, z, a, b) = 0              ……. (2)

Involving two arbitrary constants a, b and satisfying the P.D.E. (1) is known as complete solution.

General Solution of P.D.E. (1) is any arbitrary function F of given functions u, v

F (u, v) = 0                         ……..(3)

Satisfying P.D.E. (1).

Here u = u(x, y, z) and v = v(x, y, z) are known functions of x, y, z

Linear PDE

A partial differential equation is said to be linear (after rationalization and cleared of fractions) if the dependent variable z and its derivatives are of degree (power) one and products of z and its derivatives do not appear in the equation.

Quasi-linear PDE

P.D.E. Is said to be quasi-linear if degree of highest ordered derivative is one and no products of partial derivatives of the highest order are present.

Linear partial differential equation of first order

The general form of a quasi-linear partial differential equation of the first order is given as

This equation is also known as “Lagrange’s linear equation.”

If P and Q are independent of z and R is linear in z then above equation is a linear equation.

The general solution of the Lagrange’s linear P.D.E.

Pp + Qq = R      …(1)

Here P, Q and R are the functions of x, y, z and p = and q =

Is given by the equation

F (u, v) = 0

Here u = u(x, y, z), v = v(x, y, z) are specific functions of x, y, z.

Working steps to solve-

Step-1: Write down the auxiliary equation-

Step-2: Solve the auxiliary equations-

Suppose the two solutions are- u = and v =

Step-3: Then f(u, v) = 0 or u = ∅(v) is the required solution of

Example: Solve the following-

Sol:

This is a linear P.D.E. Of first order Pp + Qq = R with P = x, Q = y and R = 3z. The Lagrange’s auxiliary equations are

Integrating the first two equations , we obtain

Integrating first and the last equations, we obtain

Hence

Thus the required solution is

Method of multipliers-

Let the auxiliary equation be

L, m, n may be the constants of x, y, z then we have-

L, m, n are selected in a such a way that-

Thus

On solving this differential equation, if the solution is- u =

Similarly, choose another set of multipliers and if the second solution is v =

So that the required solution is f(u, v) = 0.

Example: Solve-

Sol:

Auxiliary equations-

Choosing multipliers as x, y, −z

Xd x + ydy − zd z = x(y + zx) + y(−1)(x + yz) – z (

Integrating

Choosing multipliers as y, x, 1, we get

Yd x + xdy + d z = y(y + zx) + x(−1)(x + yz) +

Integrating xy + z = c2

The general solution is

Example: Solve-

Sol.

We have-

Then the auxiliary equations are-

Consider first two equations only-

On integrating

…….. (2)

Now consider last two equations-

On integrating we get-

…………… (3)

From equation (2) and (3)-

Example: Find the general solution of-

Sol. The auxiliary simultaneous equations are-

……….. (1)

Using multipliers x, y, z we get-

Each term of (1) is equals to-

Xdx + ydy + zdz=0

On integrating-

………… (2)

Again equation (1) can be written as-

Or

………….. (3)

From (2) and (3), the general solution is-

Example: Solve-

Sol:

Auxiliary equations are

On integrating

x − y − z = c1

From first and second, we get

Which means

The general solution will be

Key takeaways

1. A partial differential equation is said to be linear (after rationalization and cleared of fractions) if the dependent variable z and its derivatives are of degree (power) one and products of z and its derivatives do not appear in the equation.
2. Lagrange’s linear equation-

1.4 Method of Separation of Variables for solving first order partial differential equations.

Method of separation of variables

In this method, we assume that the dependent variable is the product of two functions, each of which involves only one of the independent variables. So to ordinary differential equations are formed.

Example 1. Using the method of separation of variables, solve

Solution. 

Let, u = X(x). T (t).                     (2)

Where X is a function of x only and T is a function of t only.

Putting the value of u in (1), we get

(a) 

On integration log X = cx + log a = log

(b)

On integration

Putting the value of X and T in (2) we have

But,

i.e.

Putting the value of a b and c in (3) we have

Which is the required solution.

Example 2. Use the method of separation of variables to solve equation

Given that v = 0 when t→ as well as v =0 at x = 0 and x = 1.

Solution.

Let us assume that v = XT where X is a function of x only and T that of t only

Substituting these values in (1), we get

Let each side of (2) equal to a constant

Solving (3) and (4) we have

Putting x = 0, v = 0 in (5) we get

On putting the value of in (5) we get

Again putting x = l, v= 0 in (6) we get

Since cannot be zero.

Inputting the value of p in (6) it becomes

Hence,

This equation satisfies the given condition for all integral values of n. Hence taking n = 1, 2, 3,… the most general solution is

Example: Using the method of separation of variables, solve Where

Solution:

Assume the given solution

Substituting in the given equation, we have

Solving (i)

From (ii)

Thus

Now,

Substituting these values in (iii) we get

Which is the required solution

References:

1. Erwin Kreyszig, Advanced Engineering Mathematics, 9thEdition, John Wiley & Sons, 2006.

2. Advanced engineering mathematics, by HK Dass

3. Differential equations, Shepley L. Ross, Willey India.

4. Partial differential equations, Phoolan Prasad, Renuka ravindran, John Willey & Sons