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Mathematics III (PDE, Probability & Statistics)

Unit – 1

Definition of Partial Differential Equations


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: 

    2.     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

  •  Every dependent variable and every derivatives present in the equation is only of degree one.
  • 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.

  • Write the given family of curves.
  • Differentiate n times to get n equations.
  • 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

     


    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.

     

    The differential equation of first order and first degree is namely:

  • and
  • Example 1. Solve (UP,II 2008, U.P.B Pharm (C.O.)2005)

    Solution. We have,

    Seperating the variables we get

    (sin y + y cos y ) dy ={ x (2 log x +1} dx

    Integrating both the sides we get

    Example 2. Solve the differential equation

                      (A.M.I.E.T.E. Winter 2003)

    Solution.   

    Put,

     


     

    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 for4m

    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             (Kottayam, 2005)

    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

     


    Linear Homogenous Partial Differential Equations Of Nth Order With Constant Coefficients

    An equation of the type

    Is called a homogenous linear partial differential equation of nth order with constant coefficients

    It is called homogenous because all the terms contain derivatives of the same order.

    Putting,

    Rules for finding the particular integral

    Given partial differential equation is

    (i)                When F (x, y)=

        [Put D = a, D’ =b]

    (ii)              When  F (x, y)=

    (iii)           When F (x, y)=

    in ascending power of D or D’ and operate on term by term.

    (iv)            When = Any function F (x, y)

    Resolve into partial fractions

    Considering f(D, D’) as a function of D alone

    Where c is replaced by y + mx after integration

    Case 1. When R.H.S =

    Example 1. Solve

    Solution.

    Given equation in symbolic form is

    (D3-3D2 D'+4D'3)z=ex+2y

     

    It’s A.E. is where m = -1,2, 2

    Put, D=1,D’=2 

    Hence complete solution is

    Case II. When R.H.S =

    Example 2. Solve

    Solution.

    Putting

    A.E. is

    Put

    C.F. is

    General solution is

    Example 3. Solve

    A.E. is

    C.F. =

    It is a case of failure.

    Now,

    Case III. When R.H.S. =

    Example 4. Find the general integral of the equation

    Solution.

    With the given equation can be written in the form 

    Writing D = m and D’=1the auxiliary equation is

    Hence the complete solution is

    Case IV. When R.H.S. = Any Fraction

    Example 5. Solve

    Solution.

    A.E. is

     Hence complete solution is

    Non Homogeneous Linear equations

    The linear differential equations which are not homogeneous are called non Homogeneous Linear equations.

    For example

    Its solution,

    Complementary function : let the non Homogeneous equation be

    The Lagrange’s subsidiary equations are

    From first two relations we have, -mdx = dy

    And some first and third relation,

    From (1) and (2) we have

    Similarly the solution of

    Example. Solve

    Solution. Equation is written as

    Hence the solution is

    Particular integral

    Case 1.

    Example.

    The complementary function is

    Hence the complete solution is

    Case 2.

    Example. Solve

    Solution.

    Hence the solution is

    Case 3.

    Example. Solve

    Solution.

    Hence the complete solution is

     


    Linear differential equation of second order with constant coefficients

    The general form of the linear differential equation of second order is

    Where P and Q are constants and R is a function of x or constant.

    Differential operator

    Symbol D stands for the operation of differential i.e.

    stands for the operation of integration.

    stands for the operation of integration twice.

    can be written in the operator form.

    Complete solution - Complementary function + Particular integral

    Let us consider a linear differential equation of the first order

    Its solution is

    Where,

    (i)Now differentiating with respect to x we get

    Which shows that y = cu is the solution of

    (ii) Differentiating with respect to x, we get

    hich shows that y = v is the solution of

    Solution of the differential equation (1) is (2) consisting of two parts i.e. cu and v. cu is the solution of the differential equation whose right hand side is zero. cu is known as complementary function. Second part of (2) is v free from any arbitrary constant and is known as particular integral.

    Complete solution = complementary function + particular integral

    Method for finding the complementary function

    (1)  In finding the complementary function, R.H.S of the given equation is replaced by zero.

    (2)  Let

     

    Putting the values of y,   (1) then

    It is called Auxiliary equation.

    (3)  Solve the auxiliary equation.

    Case 1. Roots, real and different.

    If are the roots then the C.F. is

    Equation (1) can be written as

    Replacing

    From (3)

    This is the linear differential equation.

    Example 1. Solve

    Solution. Given equation can be written as

    Here auxiliary equation is

    Hence the required solution is

    Example 2. Solve.

    Solution. Given equation can be written as

    A.E. is

    Hence the required solution is

    Example 3. Solve

    Solution. Here the auxiliary equation is

    Its root are

    The complementary function is

    On putting y = 2 and x = 0 in (1) we get

    2 =A

    On putting A = 2 in (1) we have

    On Differentiating (2) we get

    But,

    On putting x = 0, we get

    (2) becomes,

     


    PDE’s are usually specified through a set of boundary or initial conditions. A boundary condition expresses the behaviour of a function on the boundary (border) of its area of definition. An initial condition is like a boundary condition, but then for the time-direction. Not all boundary conditions allow for solutions, but usually the physics suggests what makes sense. Let me remind you of the situation for ordinary differential equations, one you should all be familiar with, a particle under the influence of a constant force.

    which leads to

    , and

    These are linear initial conditions (linear since they only involve xx and its derivatives linearly), which have at most a first derivative in them. This one order difference between boundary condition and equation persists to PDE’s. It is kind of obviously that since the equation already involves that derivative, we can not specify the same derivative in a different equation.

    A solution of a PDE in some region R of the space of the independent variables is a function that has all the Partial derivatives appearing in the partial differential equation in some domain D containing R, and satisfies the partial differential equation everywhere in R.

    Often one merely requires that the function is continuous on the boundary of R, has doors derivatives in the interior of R, and satisfies the partial differential equation in the interior of R. Letting R lie in D simplifies the situation regarding derivatives on the boundary of R, which is the same on the boundary as it is in the interior of R.

    In general, the totality of solutions of a partial differential equation is very large. For example the functions

    Which are entirely different from each other, are solutions of (3), as you may verify. We shall see later that the unique solution of a partial differential equation corresponding to a given physical problem will be obtained by the use of additional conditions arising from the problem. For instance this may be the condition that the solution u assume given values on the boundary of the region R (boundary conditions). Or when time t is one of the variables, u  may be prescribed at t =0 (initial conditions)

     


    The method of d'Alembert provides a solution to the one-dimensional wave equation

    that models vibrations of a string.

    The general solution can be obtained by introducing new variables  and , and applying the chain rule to obtain

    Using (4) and (5) to compute the left and right sides of (3) then gives

    respectively, so plugging in and expanding then gives

    This partial differential equation has general solution

    where f and g are arbitrary functions, with f representing a right-traveling wave and g a left-traveling wave.

    The initial value problem for a string located at position  as a function of distance along the string x and vertical speed  can be found as follows. From the initial condition and (12),

    Taking the derivative with respect to t then gives

    and integrating gives

    Solving (13) and (16) simultaneously for f and g immediately gives

    so plugging these into (13) then gives the solution to the wave equation with specified initial conditions as

    Example. Find the deflection of a vibrating string of unit length having fixed ends with initial velocity zero and initial deflection f (x) =k (sinx –sin2x)

    Solution. By d’Alembert’s method, the solution is

    i.e., the given boundary corrections are satisfied.

     


    One dimensional heat flow

    Let heat flow along a bar of uniform cross section in the direction perpendicular to the cross section. Take one end of the bar as origin and the direction of the heat flow is along x axis.

    Let the temperature of the bar at any time t at a point x distance from the origin be u(x,t). Then the equation of one-dimensional heat flow is

    Example 1. A rod of length 1 with insulated sides is initially at a uniform temperature u. Its ends are suddenly cooled to 0° Celsius and are kept at that temperature. Prove that the temperature function u (x, t) is given by

    Where is determined from the equation.

    Solution. Let the equation for the conduction of heat be

    Let us assume that u = XT, where X is a function of x alone and T that of t alone.

    Substituting these values in (1) we get

    i.e.

    Let each side be equal to a constant

    And

    Solving (3) and (4) we have

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

    (5) becomes

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

    Hence (6) becomes

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

    By initial conditions

     

    Example 2. Find the solution of

    For which u ( 0, t) = u (l.t) =0 =sin bi method of variable separable.

    Solution.

    In example 10 the given equation was

    On comparing (1) and (2) we get

    Thus solution of (1) is

    On putting x =0

     u =0 in (3) we get

    (3) reduced to

    On putting x = l and u =0 in (4) we get

    Now (4) is reduced to

    On putting t = 0, u =

    This equation will be satisfied if

    On putting the values of and n in (5) we have

    Example 3. The ends A and B of a rod 20 cm long having the temperature at 30 degree Celsius and at 80 degree Celsius until steady state prevails. The temperature of the ends are changed to 40 degree Celsius and 60 degree Celsius respectively. Find the temperature distribution in the rod at time t.

    Solution. The initial temperature distribution in the rod is

     

    And the final distribution (i.e. steady state) is

    To get u in the intermediate period, reckoning time from the instant when the end temperature were changed we assumed

    Where is the steady state temperature distribution in the rod (i.e. temperature after a sufficiently long time) and is the transient temperature distribution which tends to zero as t increases.

    Thus,

    Now satisfies the one dimensional heat flow equation

    Hence u is of the form

    Since

    Hence

    Using the initial condition i.e.

    Putting this value of n (1), we get

     


    Q.1] Determine the solution of one dimensional heat Equation

      

    Under the boundary condition and   being the length of rod

     The equation of heat flow is

     Since the temp of two ends of rod are zero its solution is of form

     

     Using initial condition

    i]  when we get

      

      

    ii] when

     where 

    Hence the general solution is

     We get solution as

     

    Equation of vibrating string

    Consider an elastic string tightly stretched between two points O and A. Let O be the origin and OA as x axis. On giving a small displacement to the string perpendicular to its length (parallel to the y axis). Let y be the displacement at the point P (x, y) at any time. The wave equation

    Example. Obtain the solution of the wave equation

    Using the method of separation of variables.

    Solution.

    Let y = XT where X is a function of x only and T is a function of t only.

    Since T and X are functions of a single variable only.

    Substituting these values in the given equation we get

    By separating the variables we get

    (Each side is constant since the variables x and y are independent)

    Auxiliary equations are

    Case 1. If k>0

    Case 2. If k<0

    Case 3. If k =0

    These are the three cases depending upon the particular problems. Hare we are dealing with wave motion (k<0)

    Example. Find the solution of the wave equation

    Such that is a constant) when x = 1 and y = 0 when x =0

    Solution.

    Put y = 0, when x = 0

    (2) is reduced to

    Put when x=1

    Equating the coefficient of sin and cos on both sides

     

    Example. The vibrations of an elastic string is governed by the partial differential equation

    The length of the string is π and the ends are fixed. The initial velocity is zero and the initial deflection is u (x, 0)=2 (sinx + sin3x). Find the deflection u (x,t) of the vibrating string for t≥0.

    Solution.

    On putting x =0, u = 0 in (1) we get

    On putting in (1) we get

    On putting and u =0 in (2) we have

    on substituting the value of p in (2) we get

    on differentiating (3(,w.r.t t we get

    On putting in (4) we have

    On putting

    Given u (x, 0) = 2 (sin x+ sin3x)

    On putting t = 0 in (5) we have

    On substituting the value of

     


     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

     


    The axis of Laplace plane is coplanar with, and between,

    (a) The polar axis of the parent planet's spin, and

    (b) The orbital axis of the parent planet's orbit around the Sun.

    The Laplace plane arises because the equatorial oblateness of the parent planet tends to cause the orbit of the satellite to precess around the polar axis of the parent planet's equatorial plane, while the solar perturbations tend to cause the orbit of the satellite to precess around the polar axis of the parent planet's orbital plane around the Sun. The two effects acting together result in an intermediate position for the reference axis for the satellite orbit's precession.

     

    The mean plane occupied by the orbit of a satellite during a precession cycle; the plane normal to the orbital precession pole of the satellite.

     

    Tilting_of_Laplace_Plane_by_Planet_Nine.png

    1-s2.0-S027311771400088X-gr1.jpg

    In effect, this is the plane normal to the orbital precession pole of the satellite. It is a kind of "average orbital plane" of the satellite, around which the instantaneous orbital plane of the satellite precesses, and to which it has a constant additional inclination.

    In most cases, the Laplace plane is very close to the equatorial plane of its primary planet (if the satellite is very close to its planet) or to the plane of the primary planet's orbit around the Sun (if the satellite is far away from its planet). This is because the strength of the planet's perturbation on the satellite's orbit is much stronger for orbits close to the planet, but drops below the strength of the Sun's perturbation for orbits farther away.

    Examples of satellites whose Laplace plane is close to their planet's equatorial plane include the satellites of Mars and the inner satellites of the giant planets. Examples of satellites whose Laplace plane is close to their planet's orbital plane include Earth's Moon and the outer satellites of the giant planets. Some satellites, such as Saturn's Iapetus, are situated in the transitional zone and have Laplace planes that are midway between their planet's equatorial plane and the plane of its solar orbit.

    So the varying positions of the Laplace plane at varying distances from the primary planet can be pictured as putting together a warped or non-planar surface, which may be pictured as a series of concentric rings whose orientation in space is variable: the innermost rings are near the equatorial plane of rotation and oblateness of the planet, and the outermost rings near its solar orbital plane. Also, in some cases, larger satellites of a planet (such as Neptune's Triton) can affect the Laplace planes of smaller satellites orbiting the same planet.

     


    Spherical polar coordinates

    In spherical polar coordinates, the coordinates are r, where r is the distance from the origin, is the angle from the polar direction (on the earth, colatitude, which is 90° - latitude), and the azimuthal angle (longitude). It is customary to align the polar direction with the cartesian co-ordinate z and to measure from a zero ( our Greenwich meridian) along the +x direction, with the direction of such that the +y direction is at

    Therefore, points with a given value of r lie on a sphere of radius r centred at the origin and points of given lie on a cone with vertex at the origin, axis in the z direction and an opening angle of rotation . Points of given lie on a half plane which extends from the polar axis to infinity in the direction given by . In order for co-ordinate sets and arbitrary spatial points to be unambiguously related, we need to restrict angle of r to , with in the range and  within

    Untitled.png

    The equations connecting the two sets of coordinates are

     

    Notice that equation gives formulas for   rather than for . When we convert we must use the principal value of so as to obtain a result to within the range . For  we must be even more careful, as a range for the principal value of is only of length π while the range of is of length 2π. We must choose the value of that is in the azimuthal quadrant consistent with the individual signs of x and y.

    Circular cylindrical coordinates

    Circular cylinder coordinates use the plane polar coordinates and (in place of x and y) and the z Cartesian coordinates. The variable is the distance of co-ordinate point from the z. Cartesian axis, and is its azimuthal angle. The range of these coordinates are , and of course . Thus, points of given lie on a cylinder about the z axis of radius , points of given lie on a half plane extending from the entire z axis to infinity in the direction and points of given z lie on the plane with that value of z.

     

    Untitled 4.png

    The conversion formulas between circular cylindrical and cartesian coordinates are

    The co-ordinate has the same definition as for spherical polar coordinates, and just as there, it must be identified as the value of consistent with the individual signs of x and y. We have used (not r) as the co-ordinate denoting distance from the z axis, we are reserving r to always mean distance from the origin and will reserve for the axle distance defined here.

     


    Bessel functions

    Bessel equation

    The differential equation

    Is called the Bessel Differential equation and particular solutions of the equation are called Bessel function of order n.

    General solution of Bessel’s equation is

    Solution of Bessel’s equation

    Let

    So that,

    Substituting these values in the equation we have

    Equating the coefficient of to zero, we get  

    ,                      (r=0)

    ,                          

    Equating the coefficient of       (r=1)

    Equating the coefficient of to zero, to find the relation in uccessive coefficient we get

    Therefore,

    If r=0,

    If r = 2, and so on

    On substituting the values of the coefficients in (2) we have

    Fo m = n

    Where is an arbitrary constant

    For m = -n

    Bessel functions

    The Bessel Function is

    Solution of (1) is

    Where is an arbitrary constant

    If

    The above equation is called Bessel function denoted by

    Thus,

    If n = 0,

    If n= 1,

    We draw the graphs of these btwo functions. Both the functions are oscillary with a varrying period aqnd a decreasing amplitude.

    Replace n by –n in (2), we get

    Example 1. Prove that , where n is a positive integer.

    Solution.

    \On putting r = n + k

    Example 2, Prove that

    Solution.  We know that

    Substituting in (1) we obtain

    Recurrence Formula

    Formula I.

     

    Formula II,

    Formula III.

    Formula IV.

    Formula V.

    Formula VI.

    Example. Prove that where is the Bessel function of first kind.

    Solution. By recurrence formula II

    On putting n = 2, in (1) we have 

    By recurrence formula I

    From (1) and (3) we have

    On putting n = 1,

    Putting the value of from (4) in (2) we get

    Example. Prove that

    Solution. We know that

    Integrating above relation we get

    On taking n = 2 in (1) we have

    Again,

    Putting the value of from (2) in (3) we get

    On using (1) again when n = 1

    Hence,

    Legendre functions

    Legendre’s Equation

    The differential equation is known as Legendre’s equation. The above equation can also be written as

    Legendre’s Polynomial

    Definition.

    The Legendre’s Equation is

    The solution of the above equation in the series of descending powers of x is

    Where is an arbitrary constant.

    Now if n is a positive integer and

    is called the Legendre’s function of the first kind.

    Legendre’s Function of the second kind i.e.

    Another solution of Legendre equation

    Where n is a positive integer.

    If we take,

    The above equation is called so that

    The series for is a non- terminating series.

    General solution of Legendre’s Solution

    Since and are two independent solutions for Legendre’s Equation therefore the most general solution of Legendre’s equation is

    Where A and B are two arbitrary constants.

    Example 1. Let be the Legendre polynomial of degree n. Show that for any function f (x), for which the nth derivative is continuous.

     Integrating by parts, we have

    Again integrating by parts, we have

     

    Integrating (n-2) times, by parts we get

    Legendre Polynomial

    If n = 0,

    If n = 1,

    If n = 2,

    Similarly,

    Where, if n even.

    if n is odd.

    Either is in the last term.

    (n is even)

    (n is odd)

    Example. Express in terms of Legendre Polynomials,

    Solution. Let

     

    Equating the coefficients of like powers of x we have

    Putting the values of a, b, c, d in (1) we get

     


    The 1 D diffusion equation

    The famous diffusion equation also known as heat equation, reads

    Where u (x, t) is the unknown function to be solved for, x is a co-ordinate in space, and t is time. The coefficient is the diffusion coefficient and determines how fast u changes in time. Quick short form for the diffusion equation is

    Compare to the wave equation which looks very similar but the diffusion equation features solutions that are very different from those of the wave equation. Also the diffusion equation makes quite different demands to the numerical methods.

    Typical diffusion problems may experience rapid change in the very beginning but then the evolution of u becomes slower and slower. The solution is usually very smooth and after sometime one cannot recognise the initial shape of u. This is in sharp contrast to solutions of the wave equation where the initial shape is preserved – the solution is basically a moving initial condition. The standard wave equation has solutions that propagates with speed c forever, without changing shape, while the diffusion equation converges to a stationary solution. In this limit, and is governed by . This stationary limit of the diffusion equation is called the Laplace Equation and rises in a very wide range of applications throughout the sciences.

    It is possible to solve for u (x, t) using a explicit scheme, but the time step restrictions soon become much less favourable 10 for an explicit scheme for the wave equation. And of more importance, since the solution u of the diffusion equation is very smooth and changes slowly, small x steps are not convenient and not required by accuracy as the diffusion process converges to a stationary state.

    Example. Solve the equation with boundary conditions u (x ,0) = 3 sin nπx, u (0 ,t) and u (1, t) =0, where 0 < x < 1, t > 0.

    Solution. The solution of the equation

    When x = 0,

    (Ii) becomes

    When x = 1,

    Ie., P = nπ

    (iii) reduces to

    Thus the general solution of (i) is

    When t=0, 3 sin nπx = u (0, t) =

    Comparing both sides

    Hence from (iv), the desired solution is

    Example.  Solve the differential equation for the conduction of the heat along a road with out radiation subject to the following conditions

    (i)                u is not infinite for

    (ii)              for x = 0 and x = l,

    (iii)           u = lx

    Solution. Substituting u =X(x) T(t) in the given equation we get

    Their solutions are 

    If is changed to the solutions are

    If , the solutions are

    In (3),  for therefore, u also → i.e. , the given condition (i) is not satisfied so we reject the solution (3) while (2) and (4), satisfy the condition.

    Applying the condition (ii) to (4) we get

    From (2)

    Applying the condition (ii) we get

    Thus the general solution being the sum of (5) and (6) is

    Now using the condition (iii), we get

    This being the expansion of as a half range cosine in (0,l), we get

    Hence taking n = 2m, the required solution is

     

    Example. The ends A and B of a rod 20 cm long have the temperature at 30 degree Celsius and 80 degree Celsius and 10 study state prevails. The temperature of the ends are changed to 40 degree Celsius and 60 degree Celsius respectively. Find the temperature distribution in the rod at time t.

    Solution. Let the heat equation be

    In steady state condition, u is independent of time and depends on x only, (i) reduces to

    Its solution is u = a + bx

    Since u = 30 for x = 0 and u = 80 for x = 20, therefore a= 30, b = (80-30)/20=5/2

    Thus the initial conditions are expressed by

    The boundary conditions are

    Using (ii) the steady state temperature is

    To find the temperature u in the intermediate period,

    Where is the steady state temperature distribution of the form (iv) and is the transition temperature distribution which decreases to zero as t increases.

    Since satisfies one dimensional heat equation

    (iv)            reduces to

    Thus (vi) becomes

    Using (iii)

    Where

    Hence from (vii) the desired solution is

    References

     

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

    2. N.P. Bali and Manish Goyal, A text book of Engineering Mathematics, Laxmi Publications.

    3. P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Probability Theory, Universal Book Stall.

    4. S. Ross, A First Course in Probability, 6th Ed., Pearson Education India,2002.

     

     


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